give chase

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See: hunt
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Even if the intruder were to try to kick or throw the ball away, the device would simply return to give chase.
Police were reportedly investigating whether the dog may have either pulled Dion into the vehicle's path or slipped its lead and prompted him to give chase.
The police give chase to stolen vehicles, so what is the difference here?
Control room managers should take charge and drivers should have less discretion to give chase.