While that is a nice theory, things like this don't tend to really happen in empires. No matter where a roman legion fought, it would always be organised and equipped like every other roman legion everywhere else, with the differences -if there were any- being made up by foederatii. And the same goes for almost every single empire that ever existed.
Geographical and regional differences don't matter at all for the strategies employed by a large empire in different parts of its own area, as confusing as it may sound. The main objective actually is to develope a military strategy that works somewhat well everywhere instead of working perfectly on one front.
The reason for that is that a large empire must be able to react to threats on all of its borders rapidly. As such, it isn't feasable to specialize each of its armies for the area it is supposed to operate in, since said army could be needed on the complete other side of the empire just a few weeks later, not to mention that recruits currently being trained in one form of combat won't be able to be rapidly retrained for another if they're suddenly needed elsewhere.
Can't also stress it enough that homogenity in the armies of an empire helps soldiers to swiftly get used to whatever front they're starioned on since the army there will work in the same way it did where they came from, and that uniformity helps to establish monolithic power.
Your thinking of high specialization being prevalent is, in essence, to modern.