My children (son especially) has this problem occasionally. I discovered it happens when he licks his lips too much. It's a bad cycle because he'll lick his lips, causing them to dry out and hurt, then he'll lick them more to make them feel better, so they don't heal very well. I found the flavored chapstick is no good because he licks it to taste it, and then dry lips cycle continues.
Now I use either a bit of vaseline, or a nice creamy balm (mine's called "Labello")--something with no taste at all. Also, I give him a talk about not licking his lips so they will heal--it makes him more aware of what he's doing. If I can keep up the lipbalm, and stop him from licking, for a few days, the problem clears up. Hope you find something that works for you.