By airline/train or UKBF staff? There’s a difference as airport staff won’t check visa validity etc or do the entry checks that are the subject of this chain.

UKBF checks passports on departure at the Eurostar. At manchester airport at least, there are no routine outbound passport checks by UKBF

No, that's actually French border control. On the Eurostar, you clear the destination border when you check in, so that when you arrive you simply walk out.

You pass both, there are two desks. U.K. border first, then European

Not on the way out. Just Schengen checks in London (same on eurotunnel). And those stupid security checks.

On travelling to the continent by boat (hook of holland) checks at the far end. Eurostar checks at re near end. Flying checks at the far end.