I had tried to use a string with various weighted objects attached
including a plumb bob, but they are very clumsy to move around.

What I have done a couple of times already is tie a small nut (3/8") to the
end of a string line. It fits through small holes, and offers just enough
weight to pull the string down. I slowly let the string drop, jiggling it
up and down a few times, or swinging it side to side, if it gets stuck on a
blob of plaster or something, until it reaches the bottom. It doesn't have
to line up with the hole below, just get fairly close. I then take a piece
of wire with a little hook bent into the end, to reach up through the
bottom hole and fish around in the wall to catch the string. Once I have
the string out, I can tape it to the actual electrical cable and pull it up
through the wall.

It works fine as long as there's no blocking in the wall.

Anthony