I would like to get some advise on how to convince my supervisor that moving to a lab abroad, continuing to be supervised by him as well, would be better for everyone. Do you have any hints?
Maybe start with writing down the pros about moving to this lab abroad.
answer the question
Why do you want to move labs? better equipment?
what benefits will you get by moving to the lab?
do a pros and cons list.
what problems could occur by moving to the lab?
try to see it from your supervisor point of view, he might be worried that if you're abroad he wont be able to supervise you well. and who will be your intermediate supervisor on a daily basis.