Generally, the larger departments, like Dallas, will handle their own investigations of officers. They might request assistance from another agency if they have a reason to think their investigation is compromised or serious corruption is involved. In the case of corruption, a department will ask either the Rangers or FBI for assistance. The deciding factor is almost always if the corruption involves drugs, in which case they go with the FBI to get federal time. Anything else will almost always be handled by the Rangers. This is because, again as a general rule, most police officers (including the chiefs) don't really like the FBI or trust them. Smaller departments generally ask the Rangers for help with shootings and any serious offense involving officers.
One of the reasons that cops don't like the FBI is that the FBI goes after cops. A second reason is the way they step in and take over cases that the local officers have handled.
But the FBI will almost always step in when they think there is a civil rights violation. They can be asked in by the department, local politicians, the public, or anyone. They have been known to step in when they see a commotion in the news. I believe that this case is being investigated by the FBI on their own just because the news releases don't mention anyone asking for them and politicians will almost always take the credit for asking for that type of investigation. And it doesn't really mean anything to the case, IMO, since there are no allegations of anything being race related or departmental.