I would say that "handling" and "deal with" have a very similar meaning. The differences would be, "Handling" is more formal and polite and "deal with" is less formal and more colloquial for a given situation. Egs. "Can you please handle that difficult customer?" vs "Can you please deal with that difficult customer?" "Follow up" means to check on the progress/situation of something at a given point in time. Eg. "Can you please handle that sales lead and then follow up in two weeks time?"

If there was a problem, and a person is able to solve the problem, you would say "Thank you for handling this matter." You could also say "Thank you for dealing with this matter". After the person takes care of the situation, and followed up on the problem until it was solved, you would say "Thank you for your follow up action". Or thank you for following up on this situation."

A good dictionary can handle this question. Dealing with this question requires a dictionary. You can follow up on my answers by using a dictionary. www.thefreedictionary.com