Can we track matter through time by looking at different depths in space? If we look back far enough we can see all the origins of the universe, so is it possible, even if not feasible, that we could trace the history of some matter as it moves through space-time? I want to understand how looking at different depths in space and time are correlated in regards to the matter being observed. 

For example would it be possible to look deep into a certain part of space and time to find some galaxy that contributed to the matter that makes up the Milky Way today? Then somehow follow it through space-time by looking at different depths and locations in space, and see how it came to be part of the Milky Way?
Would it be possible to look deep into a certain part of space and
  time to find some galaxy that contributed to the matter that makes up
  the Milky Way today?


No, that's not possible. If we could do that, it'd mean that the matter traveled from there to here faster than its light got here, and matter can't travel faster through space than light does.

All we can do is look at similar galaxies to the Milky Way at earlier times. And because of the expansion of space those galaxies are now even further away from us than they were when they emitted the light that we're seeing now.

Galaxies develop (mostly) in isolation from one another, apart from the occasional merger or collision between neighbouring galaxies. Intergalactic distances are fairly huge, so it takes vast amounts of time for matter to travel from one galaxy to another, and matter is mostly bound by gravity to the galaxy it's in. Galactic escape speeds are pretty high, although the occasional star does get flung out of the galaxy by cataclysmic events like supernova explosions. But even then, such rogue bodies mostly end up in intergalactic space. The odds of them ending up in a different galaxy are pretty slim.