How does Akshan, Mischievous interact with Dockside Lock-Up

Yes — Akshan can take Dockside Lock-Up, and doing so effectively blanks the gear. Lock-Up isn't attached to anything, and Deploy gear can be moved by appropriate effects (Rule 830.1.b.2), so Akshan's trigger just moves it to your Base and you control it until he leaves — there's no failed move like there is with Equipment.

The important part is location. Once Lock-Up sits in your Base, "here" means your Base, so both location abilities go dead: you can never hold at a Base, so "When you hold here, kill this to play the banished card" can never trigger, and the card it banished stays in banishment permanently. The Deploy clause ("when an opponent holds here, kill this") is likewise dead, so it also stops dying on its own. Only an effect that physically moves the gear back to a battlefield would wake it up again.

Also note that Akshan can't rescue the unit — he has no Reaction, so he can't be played while Lock-Up's "when you play this, banish a unit here" trigger is on the chain; that banish has already resolved by the time you could play him.

When Akshan later leaves the board, control reverts to the original owner, but the gear is sitting in your Base, so the next cleanup recalls it to their Base (Rule 149.3).