From my reading, burials were handled in many ways. The variables involved were,
1. Size of the battle - number of dead
2. Commander holding the field following the battle - allow defeated army to take care of their dead
3. Some battles were mobile fights - did not allow time to bury dead
4. Time of year - 95 degree heat = burials as soon as possible, winter frozen ground would delay burials
The dead had every type of treatment from being shipped home individually for burial to unmarked mass graves at battle sites. Local citizens frequently got involved by identifying the dead from letters and personal belongings and making sure the graves were marked. Many soldiers before battle would pin or tie nametags to themselves in case they did not survive. People interested still search today for burial sites of their ancestors.