Quote Originally Posted by weegee
When they first appear, do they show their entire hand (i.e. their whole army) or do units continue to trickle on the board over several turns.
I believe that it's their entire army that shows up at once. If any new units enter the board, their numbers are insignificant.

Quote Originally Posted by weegee
So I'd like to make an estimation of how far they can progress before Byz will start to turn the tide, but I want to make sure I know what I'm going up against.
It's unlikely the Byzantines will ever turn the tide; in fact, they may never commit significant forces to fighting the Horde (strategic AI can be stupid like that). However, the Horde is unlikely to conquer the whole Byzantine empire either, since it doesn't know how to properly deploy its forces for both conquest and occupation. Frequently, one big Horde army will move about illogically, leaving few soldiers (if any at all) to occupy new conquests, and opening the door to peasant revolts and loyalist uprisings. In this way the vast numbers of Horde troops can dwindle slowly over the years, while the Horde's conquests stagnate.

The one thing I'd like to remind you of is this: you have little reason to fear the Horde. If you follow the tactical advice others have given (armies of halbs, arbs, ILI, and a few cav using high ground and forests), your armies will crush the Horde without much trouble. Be prepared for some frustration when attacking, since MHA can be quite difficult to pin down, and you will face a lot of them. Also, guard the flanks of your units extra-carefully, since a cavalry-based enemy can more easily threaten you from multiple sides, and a morale collapse will be disastrous for you.

Happy Horde hunting!