"Buying" a game here on itch places it into it's own category in your my collections page, and gives you immediate access to any build that is released for that pay bracket. However, "buying" a game while it's on sale for half off will not give you access to the full price future releases, and you'll have to wait for that release to be lowered to the price you paid during the sale before you can update to it.
For example, I paid $5 while a game was on sale and got the $10 release that day, but for any release that came out that was in the $10 bracket, I had to wait until the price was lowered to $5 or the game went half off again to gain access to that $10 bracket again.
Itch is a little weird like that, but they do it for the game devs that release an update and gradually lower the pay tier, letting people that pay more have first crack at the new content.