I would say that in this context it's a variant of
catch up with.
Suppose you and I want to go a restaurant after work. I can't go
immediately, but I can go after finishing a little more work that I
have to do, so I tell you to go on ahead, and that I'll
catch up with you later.
Suppose you and I are in a competition -- a race. After a few
minutes you are ahead of me, but you start to get tired. You slow
down. Now I start to
catch up with you. The distance between us is less and less. I'm
catching up with you. When I reach you, I
have caught up with you.
In the song, it sounds to me like the singer is trying to run away from
somebody, but they are following, and if he doesn't keep moving, they
will
catch up with (
catch up on) him. They will
reach him -- which he does not seem to want. (It's possible that
this is metaphoric language for trying to forget while the unwanted
memories keep coming back stronger and stronger.)
CJ