Magic in the cosmere

[Zmaray]

Do different magic systems work on other shardworlds?

if so, would a radiant on Roshar be able to burn Lerasium and become a mistborn?

[Weltall]

As others have noted, the answer is 'it depends but generally speaking, yes'. For some system-specific details:

Metallic Arts: Can be done anywhere and can theoretically be granted to anyone via different methods. Allomancy and Feruchemy can both be granted 'mechanically' to people who don't have the sDNA for it via Southern Scadrian medallions and/or the Bands of Mourning. A way to reproduce Allomancy in a less ridiculously overpowered form is discussed as a potential future breakthrough but as far as we know it hasn't happened yet. Hemalurgy can be done anywhere provided the creator of the spike has the necessary intent and knowledge and its potential to steal all sorts of things is commented on in the Ars Arcanum. And of course anyone with lerasium could become a Mistborn.

AonDor, Dakhor, ChayShan, Forgery, Bloodsealing: Because the Dor is located in the Cognitive rather than the Spiritual Realm, Selish magic is linked to geography and gets weaker the further away you get from the region it's associated with. There are ways to work around this limitation but we don't know what they are yet aside from the statement that the Moon Scepter would make such a task 'less daunting'. We do have evidence that the passive effects of being an Elantrian can be made to work offworld but we've yet to see AonDor (or any other Selish system) used offworld, that we know of. Brandon has however made comments on what a Seon bond would look like on Roshar which are interesting. We don't know enough about the process of Initiation to say whether offworlders could access these systems.

BioChroma: Breath endows you with the ability to perform Awakening simply by having it, so anyone with Breath can take full advantage of BioChroma's powers whether they're native to Nalthis or not. We see Hoid enjoying the passive benefits of Breath in Stormlight Archive when he comments on having obtained perfect pitch. We also know that Stormlight can be substituted for Breath vis a vis keeping a Returned alive and it's theoretically possible to use Stormlight to fuel Awakening but it hasn't been done yet. Brandon has said that it would be 'really hard' to become a Returned without being from Nalthis so that at least is much more restricted.

Surgebinding: Anyone with an Honorblade can obtain the linked Surges and Stormlight healing factor and anyone who successfully attracts a spren could theoretically become a Surgebinder. However, actually leaving Roshar would be a bit more difficult (some special knowledge is implied to be necessary to get your spren away from Roshar's Physical/Cognitive 'zone') and you'd need a store of Stormlight to actually perform Surgebinding. This could presumably be hacked but we haven't seen any examples of that yet either.

Sand Mastery: Genetically linked and so far as we know right now there isn't any way to get access to it aside from being born with the sDNA for it or hemalurgy. That said, there's no reason to assume it couldn't be done offworld by someone with the talent as long as you have Invested sand handy. One of Mraize's trophies is almost certainly Invested sand so we're assuming that the stuff can be taken offworld quite easily.

Shades: The Ire are evidently concerned about these in some way because they have a means of detecting their presence and if we can take the Bands of Mourning broadsheet at something approaching face value, it's possible to use Shades in some way to power technology. The two may be linked, if the Ire's Shade-scanner was meant to detect Nazh's actions. In any case, it's a manifestation of Investiture with offworld application.

Aviar: We know that birds not native to the Pantheon can gain talents but we don't know if other types of animals could, including animals from other worlds. There's no particular reason to assume that a talent once granted would not work offworld.

[+Extesian]

So to add to the great answers from Weltall and Pagerunner, there's a really good WoB explaining this, I'm spoiler for size. 

Spoiler

QUESTION

So we know that you can’t just have someone--if someone were to do something similar to Hoid, he can’t just pop and go ‘oh look, I can now do Allomancy or now do Surgebinding’. What about Breath? Could somebody give Breath--could they still get the benefits--

BRANDON SANDERSON

Oh, good question. Yes you can, actually. Breath, once it is given to you, it is being keyed to you--your Identity. So that transfer makes it yours to use however you want.

QUESTION

So you could Awaken?

BRANDON SANDERSON

You could Awaken. If you were to somehow make it there, you would be able to Awaken. It’s the easiest of magics to get the magic from, and then to manipulate. Because it has keyed into it Identity.

QUESTION

[garbled]

BRANDON SANDERSON

Yes, you can take Breath onto another world. In fact, you’ve seen characters do this.

QUESTION

[garbled]

BRANDON SANDERSON

Yes, it would work the same way. The only magic that is location-dependent-- the ones who aren’t interested in this, just hum to yourself, okay? You don’t need to know any of this stuff to enjoy the books, okay? I’ve [written] them so that each series can be read independently, and enjoyed. There is behind the scenes stuff, and if you want to dig, it goes pretty deep. So on Sel, we have AonDor. AonDor is based on the fact that the Dor, which is an amalgamation of Dominion and Devotion, has been pressed together and stuffed into the Cognitive Realm by Odium who didn’t want it to gain sentience, as Investiture will do if it is left alone. It will either seek someone to be its Vessel or it will gain sentience. He pressed it in there; he pressed it together, which creates the violent reaction, because those two Intents are opposed. And that is the foundation of the magic. Because it’s stuck in the Cognitive Realm rather than the Spiritual Realm (the Spiritual Realm is location-independent; Cognitive Realm is location-dependent), it makes the magic on Sel only work in close proximity to what is keyed through there to the location they’re keyed to. This has to do with Identity and Connection--mostly Connection. So that means you can’t do AonDor on another planet, but you can do other magics works anywhere, because they’re drawing the magics specifically through either the place, or they’re end-neutral, like Breath is, and don’t need any extra power.

"but you can do other magics works anywhere, because they’re drawing the magics specifically through either the place, or they’re end-neutral, like Breath is, and don’t need any extra power"

This is the easy way to understand location with Cosmere magic. If it's end-neutral, you are using your own investiture to power the magic.  Because you're not drawing from an external source or simply doesn't matter where you are. This is mainly Awakening and feruchemy.

For end-positive systems you're drawing investiture from a Shard directly (allomancyn Dor) or from investiture that came from a Shard but is captured somewhere (surgebinding, sandmastery). But even then that's not usually location dependent because Shardic investiture is generally in the spiritual realm, and the spiritual realm is location independent.  Dor is the difficulty because it's in the cognitive realm, which is location dependent, and that's why it's the only system that is not portable. 

Different systems, as has been said, have complications though. With surgebinding you actually have to take your spren offworld which is hard. But with sand mastery for example, the sand (well, the lichen on the sand) will absorb any investiture (eg can be charged with stormlight).

The difficulty with most systems is it's hard to access the system from one planet if you're from another planet, a lot of them are genetic, which means you then need to hack Connection ot Identity or your spirit web to access them. Breath is the easiest because it provides both the investiture to use and the ability to access the magic system because it attaches to your identity automatically, and because it's not 'leaky'.

That may not add anything to the other explanations but i hope it helps. 

Edited May 16, 2017 by Extesian

[Julio]

Yes, they do. Hoid uses Allomancy on Roshar, and mentions a side effect of being an Awakener. Also, he uses his Yolish lightweaving in Warbreaker. 

And... yes, most likely a radiant could. Certainly a normal person could do so... or even special people like Hoid. It is, however, possible that Allomancy and Surgebinding would mess with each other in such a way that it wouldn't work. 

[Pagerunner]

All magic needs to be powered by a Shard. In some cases, like Allomancy, there is a direct connection to the Shard in the Spiritual Realm, so it can be used anywhere. (Magic from Elantris is a rare exception, where you can only draw on the Shard's power in a certain location.) Some magic systems require you to locate a source of Investiture in the Physical Realm; Surgebinding and Sand Mastery both need some provision of Investiture (either Stormlight or Invested Sand), and when you run out you can't use that magic any more. There is the possibility of 'hacking,' of powering one magic system with another Shard's power (like Awakening with Stormlight instead of with Breath), but I don't believe we have seen anyone do that beyond Vasher consuming Stormlight instead of Breath.

The second part of your question deals with Initiation, how to access a magic system. This also varies by system; some are genetic (Allomancy, Feruchemy), some can be done by anyone (Hemalurgy, fabrials, Awakening), Surgebinding requires you to act in a manner to bond a spren, and we don't actually know what's up with AonDor yet. (It's still a mystery why some people become Elantrians.) There are some methods to 'hacking' here as well, that seem to be related to a Shard's direct essence. Lerasium can make anyone a mistborn; anyone with an Honorblade can Surgebind. So, for a Rosharan mistborn, yes, that is possible. But there are limits; you can't use Soulforging to make yourself a Surgebinder, for example.

[Oversleep]

49 minutes ago, Pagerunner said:

There is the possibility of 'hacking,' of powering one magic system with another Shard's power (like Awakening with Stormlight instead of with Breath), but I don't believe we have seen anyone do that beyond Vasher consuming Stormlight instead of Breath.

Actually our prime example of hacking magic systems is powering Feruchemy with Allomancy aka Compounding