Every relationship and everyone is different so there's no set rule in these cases about what USUALLY happens.
Some people move on quickly and therefore don't have problems being friends with their ex's because they simply do not have any romantic feelings for them anymore. That's pretty much the key - you can really only truly be friends with an ex if you don't have romantic feelings for them anymore. Otherwise, you're not really friends right? because you shouldn't have romantic feelings for your friends.
If you are in a position where you still feel jealous or possessive or get upset at his new girlfriend (or future new girlfriends), then you are not ready to be his friend. As a friend, you would be happy that he has found someone to make him happy.
Personally, I like to keep in contact with my ex's simply because they are special to me and were a part of my life so I like to know at the very least if they are still dead or alive. Facebook friends if you want to call it that - we don't meet up often and don't talk regularly but will say happy birthday and Merry Christmas once or twice a year and if I see they have posted pics up, then I may take a look or not. My recent breakup was to a guy who I truly loved (and still do), and therefore I need a bit space in order to try and get myself together again before I can even attempt to be friends with him.