If units with mobility are in my HQ. When I commit my Warlord to a planet and take the units with mobility.

At the start of combat, am I able to move the units with mobility to an adjacent planet?

And if so, Do they return with the warlord? Or do they remain were I moved them?

Mobile allows a unit to move to an adjacent. It does not require the unit to exhaust to do but it will still be at the new planet exhausted.

Units never leave with a Warlord simply because the Warlord leaves. They can retreat too and they leave the first planet when it leaves play. The units that move with the Warlord are not attached to the Warlord from then on.