In Australia the doctrine is Police do not give chase, EVER! If the suspect's car starts speeding, they have strict protocols in place to back off, radio in the situation and follow from a safe distance
Studies have shown that chasing only escalates the danger to property and the public. So the correct response, is to radio in for support and do everything to protect life and property. They can't really outrun a radio.
There was a recent, going back a few years now, chase that took five days from when they first attempted to stop a car, to the actual capture. There were gaps, but police had photos, videos, registration plates and descriptions of the suspects. They knew it was only a matter of time before the suspects would be caught.
And guess what, not a single shot was fired