It's been over 20 years since the last time an aircraft shot down another aircraft in combat. If it ever happens again, they probably will want to take a look at the crash site - mission and fuel permitting. They might even be required to do so if a parachute came out of it.
But we're talking about a high-value alien vehicle here. Part of the mission will certainly be to track where it ends up. Especially if it goes down over a populated area.