noun
pursuit
per-SOOT/pərˈsuːt/
Continued following directed toward a target.
The older runner continues in pursuit of the runner a few strides ahead.

Etymology
Following forward supplies the image of chasing people to overtake them. This extended to seeking an object or goal through continued effort, whether or not a literal chase was involved.
In context
… he started a pursuit and pulled the lights and sirens on and the car just took off from him
More examples
- They travel far in pursuit of better jobs.
- Gardening is his favorite weekend pursuit.
- The officers continue their pursuit of the fleeing driver.
Forms
pursuit · pursuits
pursue verb — Follow or seek something with continued effort.