On the active battle pass track, you can see how many points you've earned in the current tier. On inactive battle pass tracks, partially completed tiers do not show progress.
I'm fairly certain that the points transfer to the currently active path. After I finished the initiation path I had earned ~8 extra tokens, it had automatically chosen the first path but when I changed to another path straight away it showed I had 8 tokens towards the first unlock the now active path. I think you unlock a reward every 10 tokens and it just unlocks on the currently active path, so when you change to another active path you haven't actually progressed 8/10 tokens for that item and instead will just unlock the next item on whatever the currently active path is in 2 more tokens.
Ok, I can confirm this. I switched back to this inactive path after progressing another path, and the points to the next tier changed to match the other path.
