Do Constraints move from one database to another?
Yes, in SQL Server constraints are defined within each database within the table definition. So if you physically promote a database to a higher environment through backup and restore, the constraints will remain within the database. If you script out objects, the default behaviour will be for the constraints to be scripted out with the table definition. 

Microsoft's TechNet has a decent explanation:
https://technet.microsoft.com/en-us/library/ms189862(v=sql.105).aspx