renamed mobs lose their ability to wander

after you use the name tag item on any of the mobs, they'll stop their wandering completely and will remain motionless in one spot after they stop.

you could bunch up several sheep in one place after renaming them and leave them somewhere in the center of your town/base, and they'd be still there in the same position when you returned later, minus some possible slight movement from being pushed by those without name tags when they pass by.

they'll only react to things like enderman to rain, creeper to an ocelot, undead mob to sunlight or hostile mob to a player in survival mode.