Cassette movement

Cassette body isn't aligned with the axle correctly - either came that way or the body is loose.

I had the same problem when I bought a pair of new wheels, as the spacer behind the cassette was being moved out by some rivets which held the cluster together. My spacer had cutouts to accomodate the rivets, but they weren't aligned with the rivets, therefore the assembly was wobbling slightly, as seen in your video.

Edit: sorry, I was wrong. My problem was when the cassette was spinning, and not when it was stationary. The problem you're seeing here might be down to poor tolerances within the hub. Anyway, your LBS should be able to sort this.

A missing spacer or worn hub bearings/freewheel will cause cassette play; however it should not occur otherwise and that video looks like a duff freewheel!

A freewheel can be damaged by not using a chain whip to oppose torque wrench force to lock a cassette to the freewheel; however the design of chain whips seems crap because there is nothing to hold the free chain end tight to the cassette, so I fixed a bit of gear cable to mine.

My freewheels do a bit of that,especially the cheap ones, but my free hubs don't.

You're right @IHPhoto, sorry.

And @wellcoordinated sure, but not that much usually.

Come one guys have you checked yours recently? Every free hub does this to some degree. It's perfectly normal.

It cn't be a bent axle as the axle is held stationary. Either the cassette is not fixed in position or the freehub has a major bearing issue - often the outer one(s). I have had the latter develop and it often means a new freehub or at the very least new bearing/adjustment if they are of the free type. Most these days use sealed industrial types.

That does look as if the axle is bent, have seen that elsewhere.

Otherwise the cassette lock ring is loose or the bearings aren't adjusted properly or something. Take it to LBS and if not satisfied with the explanation take it somewhere else for a second opinion.

Its worth checking, if you have the tools that is, that the cassette is properly seated on the free hub. Sometimes it can jag, but not usual with a new free hub body. Hard to tell with your video whats happening. Note that the teeth on each cog do lean left right, which is intentional as it helps the chain grab when shifting between cogs.

Likely you're missing a spacer or the lock ring is not tight. Just take it back to your retailer (Evans at a guess - Trek bike n'all)
And they should sort it.

It's worth checking that:

1) cassette lockring is tight
2) more likely....the freehub needs adjusted.