Most of the fieldstone walls in northern Westchester were built dry, with no mortar, and that is why so many of them are still standing. Frost moves the ground every winter, and a dry wall moves with it and settles. A mortared wall on the same ground tends to crack.
When a section slumps, take it down to sound stone and rebuild that length dry, using the same stone. Most of what you need is usually lying at the foot of the wall.
A rebuilt section will look clean for years, and lichen is slower than most owners expect. Leave it alone and it will catch up.