What happens to the people born on Feb ... 29? do the stay one until 4 years past?

No, biologically they age just like everyone else

Explanation:

The calendar is just an invention by man to keep track of time. The reason why we have an extra day on February every four years is to make up for the extra time we have spare. If you are born on this day, it will not influence the rate you age.

Well technically yes because the birthdate that you were born on will not appear until the next four years but you have lived those past few years so you are going to be the age you would be if not born on leap year