Actually, an Exalt's power comes from a secondary soul known as their Exaltation that attaches to their human soul, giving them their divine powers. If the Exalted is removed from its home universe, its Exaltation will remain attached to it.
The thing is, you shouldn't be removing Exaltations from the Exalted continuum in the first place. There are only a few hundred Solars, Lunars, Abyssals, and Sidereals, the exact number varying for some of those Exalt types based on the time period and how active the various factions of the Primordials have been lately, and the continuum sort of needs those to deal with the massive amounts of demons, Fair Folk, Neverborn, and other such nasties that fill the continuum, not to mention the chaos-spawned creatures coming in from the Bordermarches and Middlemarches every other day.
For those, you should destroy the bearer and return the Exaltation to wherever it's held in between hosts, though I suppose one could try to find a way to remove the Celestial Essence of the Exaltation if one really wanted to recruit a specific Exalt of the limited sort.
Preferably, you should also neuralyze the Exalted first before killing it/removing the Celestial Essence to ensure that the Exaltation doesn't pass on uncanonical memories to its next host, but if you don't, it's not the biggest of deals, since Lytek clears most of the memories out anyway, and would find Suvian impulses unsuited for the next Exalt down the line.
Infernals are even less in number than the previously mentioned four, and there's the whole "over time they're going to turn into the next generation of the Yozi, a.k.a. the gigantic quasi-demonic creatures that tried to kill the world" problem with the Infernal Exalted that says it would be a Very Bad Idea to let one run around HQ unchecked.
Alchemicals and Terrestrials can be created in any number, though, and though they are linked to an Exaltation of sorts, it's less powerful, and in the case of the Alchemicals, synthetic.
I'd imagine that a Terrestrial Exalted as an Agent would be treated like any other Agent that comes from a species capable of using elemental powers, like the Benders from A:TLA or the Toa from Bionicle. They'd have access to more raw power, and would thus be likely to be transferred to the sections of the PPC that deal with more raw power, such as the Special Operations Division or the Eclectic Subdivision of Advanced Species. They might also have places in the Security Departments, because their power levels would be very useful for keeping order and combatting threats.
An Alchemical, on the other hand, is a different matter, since they're scaled-down robot versions of the other Exalt types. They'd still work well in the same places the Terrestrials would, but more specific postings would depend on their caste. The highly intelligent Orichalcum Alchemicals would be able to take up positions in research-based Departments like DMSEAR and DoSAT, the Jade Alchemicals might be suited to duty in FicPsych, Medical, or another place where their determination, high-spirited attitude, and still-capable level of power would be best put to work, so on and so forth.
These are generalizations, of course, since personality would still be in play. Maybe one of the Alchemicals is too impatient for a position that his fellow would take to gladly, or a particular eye for detail and mind for figures gets one Alchemical sent to Finance or Legal. It's really all about the individual in the end, but high levels of power mean that they should simply be put where those levels of power could do some good without showing everyone else up.