I eventually find a trick: when adding a second delay between the mixer and the output, each part routed to the mixer seem to be cached properly in time. The fun part is that this delay can be set to virtually 0 (like 0.01) for the trick to work, but not 0!
I still have a little pop when transitioning, but that’s not a timing problem anymore, it’s coming from the intro part stopping abruptly (since it’s not a real concatenating). Baking a faded out release tail at the end of the intro should definitively resolve the problem.