As other people have noted, it's very much a case-by-case thing.

There are 'verses that share the same magical system -for instance, Dragonlance and Forgotten Realms are both D&D settings, and use the same magical systems. A mage from one could reasonably be expected to have no trouble in the other. I'd be willing to accept crossovers between 'verses that have similar systems, especially when they're ill-defined: A D&D sorcerer would probably not have too many difficulties working magic in Valdemar, for instance.

On the other hand, there are 'verses that are very definitely not compatible. For instance, a Potterverse wizard isn't going to be able to do anything in Earthsea, where everything works by knowing things' True Names. That said, I wouldn't have a problem believing that someone inclined to magic in one 'verse is also inclined to magic in other ones, even if the underlying mechanism changes.

Though I've had my own take on things. I have a few Agents from 'verses other than World One, and those of them that have any real powers (one LOTR-verse Elf with his natural physical abilities, one Potterverse wizard, and one as-yet-tentatively potential Jedi Padawan) can still access their abilities on missions if not disguised.

As I see it, when the Disguise Generators put somebody into a disguise, that person is then limited to the abilities of the being they're disguised as. If they drop the disguise, they can access their own abilities again, unless the continuum has specifically disallowed said abilities.

In HQ, of course, Disguise Generators aren't necessary and powers are unblocked there.

First question: Nanohaverse (Magical Girl Lyrical Nanoha) magic is based on the so-called "Linker Core", a sort of immaterial organ that allows the mage to process magic (and it's also the "fuel tank" for mana). Since it's something inherent the character and not the continuum, it should work out of the Nanohaverse, right? (I've got an Agent from that continuum and I was thinking about getting another one)

Second question: The Agent I was going to recruit is Material-S from the MGLN Battle Of Aces PSP game. She's a magical construct "clone" of the main character, Nanoha, and looks like she can be a good Agent: she's quite calm and polite, so she shouldn't go insane and can restrain more hot-blooded Agents, and has the psicological need for blowing enemies up, so she can happily kill Sues and disintegrate Author Wraiths. She has quite too much firepower (Nanoha, the girl she is a clone of, is called "White Devil", "Gundam in schoolgirl dress", "Sailor Beamspam" and "Demonic Cannon Girl". Yeah.), but nothing I cannot nerf down (power limiters exist in the home continuum itself).
Provided she is canonically dead (and, besides, she's in a side continuum made for the game, she doesn't exist in the main MGLN canon), may I recruit her?

It depends on the powers, I think. Also, in messed up fics pretty much everything goes and canon's on your side.

For the HQ: This thing's basically a mashup of everyting, it keeps changing more often than Hogwarts stairways and operates on the base of the Laws of Narrative Comedy.

My take on it: One of my characters is a Warcraft!Paladin. She can use her powers everywhere, but they are way stronger in the Warcraft continuum.

Well, one of my agents, Lee Keaton, is able to do magic, but given that her 'Verse's magic usually isn't compatible with other 'Verses that don't have elemental-based magics, she tends not to use it outside of HQ.

is the general power level of the continuum the agents will be working in. A lot of worlds are just not survivable by a standard human, so an agent would require above human attributes to work in those worlds. Drop the same agent into a World One based continuum and the agent would be either ridiculously over-powered or constantly restricted.

There's a lot of agents in the PPC from various continua, and so far from what I've seen around the general consensus is that they have access to any race or continua specific powers they used to have, seeing as it's usually a matter of biology. For instance, Agent Gremlin in the department of Bad Slash is a meta-human from the DCU who has the power to manipulate electricity and machines. Because her power is based on a meta-gene complex, and is this a part of her biology in much the same way your arm is, she can use this power in any continuum she wishes.

However, there is an addendum when it comes to magic and magic-like powers, which is that only magic from that continuum is usable since it follows that continuum's rules. For instance, a wizard agent from Harry Potter could use his magic in a mission in the Harry Potter continuum, but if he went to the Dresden Files continuum, where they have different rules for magic, he couldn't use his H.P. magic without breaking the rules of canon, which is a Bad Thing. Since chakra and jutsu are basically another type of magic, they fall under this same rule. However, within HQ magic should work just fine.