The instantaneous change occurs when you consider the Doppler shift in only one dimension.  In three dimensions you can consider the correction when the velocity vector and the separation vector are not parallel.  Usually such corrections go like $\cos\theta$, where $\theta$ is the angle between the two vectors, but more complicated things are possible.

Years ago I sat down and computed the speeds for which acoustic Doppler shifts correspond to musical intervals.  That gave me the superpower of being able to stand on a sidewalk, listen to the WEEE-ooom as a car drove past, and say to myself “a *major* third? They're speeding!”  But because of the $\cos\theta$ dependence, the trick gets harder as you get further from the road.