Well in the old fluff (and iirc), the armour of exarchs had a kinda mini infinity circuit, since the violent nature of their souls could not be allowed to join the craftworld infinity circuit. So what happenen was that the souls of the exarchs remained within the armour, while the most powerfull (in terms of willpower, usually the first one to carry the armour) soul basically stays in control. There was not really a finite border between a Phoenix Lord and an Exarch.
When the armor got damaged (as back then the PLs where a bit like thousand sons, basically animated suits of armour) badly, another Eldar could put it on. Over time his body and mind would fuse and be absorbed by the armour, his soul joining the infinity circuit. Thats where actually much of the power of an PL came from, from the psychic might of the combined soules resting in the armour.
In more recent fluff things have changed quite a bit, the whole concept of the empty shells of armour got dropped, among other things. Still, i prefer the old interpretation of the Phoenix Lords. It shows very well, that they have sacreficed more than most living (and dead) Eldar (save those who have been taken by Slaanesh).
Now could such a process happen in battle? Without having read the book, id say it depends upon how damaged the armour was. I suspect that the armour itself can be put on relative quick. Cumbersome knightarmour that takes an hour, and the help of a squire to put on dosent sound very Eldar to me. Id doubt, that the new bearer would be instantly up to full power, as the fusion, if only of mind and soul takes time i suspect, depending upon the willpower and character of the new bearer.