The field mapping is persisted (stored) in the mxd. If you transfer the mxd to another machine and repair data sources, the field mapping may not be picked up.
During the repair data process the internal layer id may change which means the layer you now have in the map will be technically different from what was stored in the mxd. The comparison isn't done by layer name for example and attribute transfer isn't robust enough to work out its the same layer. If you dont have to repair data sources after transferring the mxd it 'should just work'.
The help doesn't describe the limitations. I'll check for and make an internal CR (Change request) to look at this issue in ArcMap. It hasn't been ported to Pro yet. I'll also enter a knowledge base article to describe the limitation of transferring an mxd with data repair.