If you choose not to chase routers, the enemy army will escape and regroup. *None* of their surviving numbers will desert, in fact some of their dead soldiers may come back as recovered casualties. In RTW the saying "a man who flees will fight another day" holds literally, leaving you with only two options to end the threat caused by enemy soldiers:

A. Destroy them
B. Bribe them

The exception is starving by siege a city where the garrison is too scared to make even the last desperate sally. That is the only occation in RWT when troops will surrender.