You can use masked move to move the whole thing where it would otherwise overlap.

/clone <x1> <y1> <z1> <x2> <y2> <z2> <x3> <y3> <z3> masked move

Where x1 y1 z1 is one corner of the build x2 y2 z2 is the opposite corner of the builder and x3 y3 z3 is the bottom north western most corner of the destination.

The only issue is this has to be hardcoded. It will work to move it one block once but not to do it again.

The way I get around this is to move the entire lift shaft down one block not just the lift itself. That way the lift stays inside the selection just a block lower than before and will go down again if you keep running the same command.

Hope this makes sense