Yes, leather climbing shoes stretch and can even become quite baggy if you choose the wrong size and they stretch a lot. Synthetic shoes barely stretch, and even if they do, its so minimal that most don't even notice a difference.
Yes, climbing shoes do tend to stretch after a few weeks of use. Synthetic climbing shoes usually stretch just about half a size at most, and many times do not stretch at all, while leather shoes normally stretch between half a size to 2 full sizes.
This should be taken into consideration when buying climbing shoes, since if you get synthetic shoes and size down 2 sizes, expecting them to stretch 2 sizes, you are in for a disappointment since they will at most stretch only about half a size. The same goes for leather shoes, if you don't take into consideration that they will stretch and don't downsize, you will end up with very loose and floppy shoes.
