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.
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.
