The DMG Says the following about dashing during a chase:
During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.
If there are vampires or vampire spawn in the chase, would this rule apply to them? RAW I don't see why not, but common sense suggests that if they don't need air then they wouldn't get exhausted from exercise in the same way a living creature would.