Lock transformation in local coordinate system I want to lock the transformation to certain axes only, but when I just lock the axes in the Transform panel, they seem to lock global coordinates, even when Transform Orientation is set to Local.



Note that locking the axes removes the arrows from the local gizmo, which makes this all the more confusing.
That's what the Transform Location locks are for, they lock the objects on the Global axis, the fact that you switch the Transformation Orientation from Global to Local won't change it.

If you want to constraint an object on its local axis, use a Limit Location constraint, switch the Convert to Local Space and choose on which axis you want it to be constrained.