Here's the rules that seem to pertain in BL and BR:
BL page 24 - "Jump Drive: If destroyed, the ship may not enter jumpspace. If not destroyed, find the percentage of hit boxes that have been marked off. Multiply the ship's standard jump performance by this number and round fractions down. This is the current jump performance number."
BL page 25 - "Bridge Destroyed: ...The ship is no longer capable of manoeuvring, entering jump,..."
BL page 30 , Scenario 1, Deep Black - "Winning: The Zhodani player wins...To get the information back, the ship must either exit the map at either end (but not the sides) with functional jump drive, or it must communicate the information to another Zhodani ship which then exits the map at either end with functional jump drive.
BL page 30, Scenario 2, Rearguard - "Winning: ...In either case, the Zhodani player must do so before the end of the 20th turn, or Bard Refuge will have had time to reach its jump point and escape.
There are other references to exiting the board and being capable of jumping in Scenario 4: Corsair!, Scenario 6: Smash and Grab, and Scenario 7: Cold Recovery.
As for Battle Rider, the 'Other Considerations' section on page 15 covers jumping in the "exiting the map" sub-section. It's the same general idea as expressed in the BL Scenarios - but with explanation as to the abstraction of having to exit from the ends of the map, not the sides.
Hope that helps.