What happens to my adwords campaigns when the budget is less than 10$ a day?

It depends... I've seen a lot of small campaigns with small budget do just fine on any setting because Google isn't fighting to spread out your terms across the entire playing field for an entire day.  However, if you've got a really large campaign, with thousands and thousands of terms, various match types, etc. etc. etc., and a low budget, then Google just doesn't know where to start and ends up doing next to nothing. Even with decent sized budgets, $5-$10k a month, I've seen Google stall out in the beginning to where you have to artificially double your budget for the first couple of days just to kick start things, then pull back to another number, that isn't always your actual budget number, but gets you your actual budget number. Using Accelerated Delivery is an option, but that can be tricky as well because you could miss some key segments of the day because you blew through your daily budget.  With that setting, I also advise using time of day targeting to make sure you're on during the money hours for your business (which may take some time to study, so you may need to test different windows throughout the day).

I recently was asked a similar question, "Why Did My AdWords Campaign Go Over It's Daily Budget?" and my reply was this 2 min video response: