Occasionally I'll have part A move when mating it to part B, even though part A's position was fully defined with other mates. For me they don't both move out of position, as you said yours does, but part A will move instead of part B. When this happens I go ahead and accept the mate, close out the mate command, then rebuild with Ctrl+Q. Both parts will then move to the correct positions.
You might check to make sure that you don't have a dismissed message that talks about making the current mate and breaking/flipping existing mates.