What happens when your dependent moves out 1/2 way through the year

It depends.  He is your qualifying child if all of the following are true:

1. He was under 19 on December 31, 2016 or under 24 and still a student (college or finishing high school).  
2. He lived with you at least 6 months and 1 day (which is reported as 7 months in TurboTax).
3. He did not provide at least half of his own support.

He is your qualifying relative if:

1. He is 19 or older (not a student in the year) or 24 or older (even if he was a student)
2. He made less than $4050 for the entire year
3. You provided at least half of his support. (Regardless of where he lived)
   

If either of the two situations is true, you may claim him as a dependent.  He may also file, but will select the option that yes, someone else will or may claim him on the return.

If you can answer yes to the first two questions on the Qualifying Child test, but no to the third, he is not your dependent, but you may still claim him for Earned Income Credit if you otherwise qualify.  You will enter his information in the dependent screen, and it will ask you if he provided half of his own support.  When you say yes, TurboTax will say you can't claim him as a dependent, but the next screen will show that you can claim him for EIC.  He will then file as a "regular" taxpayer.

If he doesn't qualify either as a qualifying child or qualifying relative, then you are unable to claim him as a dependent, and he will file his return as a "regular" taxpayer.  Please see this IRS publication link:  https://www.irs.gov/publications/p17/ch03.html, and be sure to see the information on "Qualifying Child" and "Qualifying Relative".