Rundown mechanic

[JJ]

R3, ground ball in the infield. Fielder throws home and gets R3 in a rundown. I know if the defense traps the guy off third and the first throw is there the base ump has the call on R3 diving back into third (first play by an infielder), but once R3 is in the rundown does the base ump stay involved? If he does, who pays attention to B1 touching first, being obstructed there, rounding the bag and maybe heading for second (as he is supuposed to do if R3 gets in a rundown)? Thanks -