Turned out, Seek action is a pain to use with Ping-Pong or Flip-Flop loops.
For some reason it pauses the tween, and you can't resume Ping-Pong or Flip-Flop, you can only "Resume at current progress", which resumes it for one cycle. And then you have to start it in reverse. And only then you can do PingPong..
Also, the seek scale from 0 to 1 is for forward movement only, not for forward+reverse. If you need to start from some point in reverse movement, you need to add yet another several events and conditions...