After colour matching one clip, how do you apply that same correction to all the other clips?

How to apply one colour correction to every clip

Two ways, and the second is better. You can copy the corrected clip and paste its attributes onto each destination clip. Or put an adjustment layer above the clips that need the same treatment and do the grading on that layer instead, which leaves one place to change it later.

Copying the correction clip to clip

The first answer described the direct route: select the clip that already carries the correction, copy it, then select the clip you want it on and paste attributes rather than pasting the clip. It works, and it is the obvious thing to reach for when there are two or three shots to fix.

The adjustment layer, which the thread preferred

The second reply offered the better version of the same job: place an adjustment layer on the timeline above every clip that needs the same treatment, and make the colour adjustments on that layer rather than on any clip. The correction then applies to everything underneath it.

One place to change. Revising the look means editing the layer once instead of repasting onto every clip.
Its span is the timeline. Trim the layer and you change exactly which shots it covers.
The clips stay untouched, so the original correction on any individual shot is still there and still separate.

Which to use

Pasting attributes suits a small number of shots, or shots that each need something slightly different.
An adjustment layer suits a run of clips that all need the same thing — the case in the question, where one match had to reach everything.
The two combine: correct each clip individually to bring them into line with each other, then grade the whole sequence on a layer above.
Both ideas are general. The reply described them in one editor, but adjustment layers and pasting attributes exist under some name in every major one.