How do I cut out a moving mouth?

That particular technique uses a shape mask, or a regular shape used as an Image Mask (doesn't matter which you use.) I'll describe using a regular shape applied to an Image Mask.

Start with the pen tool and draw a rough shape around your mouth on the first frame. Set some feathering if you like. Set the fill opacity to about 80% to start - you'll create the mask at the end.

Go into the Shape inspector > Geometry tab and for the Control Points, when you mouse over that line, you'll see a disclosure triangle appear on the right edge. Click on that and select Add. That's all you have to do to set keyframes.

Move the playhead to the next time - where your mouth is near or at a direction change (either opening or closing.) Select the Edit Points tool and adjust the control points around the current shape of your mouth. Move the playhead and repeat the process. Keep moving the playhead in time until you get to the end. Play your project to make sure everything is matching correctly, and if you have any problems, just move the playhead to the frames you need to make adjustments to.

Reset the fill opacity of your shape to 100%.

To the video layer, right click on the layer and Add Image Mask. Drag and drop the shape onto the image mask sublayer. Export.

depends on where you like to put it on.

To put a human mouth onto a human face asks just for some 'beef' around the mouth and color adjustments (and tracking, if the to-be-pasted face moives). Did that to make somebody sing - hilarious 🙂

In your example, human lips on doggy/furry head, I would use 'reverse green screen':

paint the area around your mouth green (don't let anybody see you while recording 😝)

The color-key in FCPX/Motion is very 'forgiven', use your kids watercolor or 'clown makeup'... some light would help.

in Motion, apply a dirt-mask to exclude face, and then apply color keying.

apply motion tracking/keyframes if needed.

gets tricky when subject rotates head.. you have to mimic that while recoridng....

Everyone has put up the same answers here as at FCP.co so I'll add mine, too: Buy a chromakey head cover, cut out the mouth.