It's important when upshifting to the next gear too.
Some people like to feel the "pull" all the way through the shift, so they are already on throttle as they let the clutch out, and they just let the sucker slip until the car catches up the RPMs. Automatics can do this, in a sense, because the hydraulics can transfer the force from one valve to the next in a buffered fashion (via accumulators) But for a Dry clutch Manual, it's just friction material and if you slip it, it won't last long.
I learned from test driving heavy dump trucks and Semis that if you throttle through the shift, you will smoke that trucks clutch in moments. Heck If you stab the throttle uphill already in gear and moving, but in too high a gear, you'll smoke it.
So you have to be almost fully off throttle (just enough to keep RPMs where they need to be at the next gear) while shifting and until the clutch is completely released. once you are fully engaged with the next gear, then you hit the throttle. So there is a moment of "coasting" in neutral while you are moving the stick, and until you release the clutch. Not the fastest way to drive but the best for the clutch.