How to Pick Up & Place in Carry On

• Pick up: look at the target block/entity and sneak + use (the default carry control, rebindable in Options > Controls). You will see the object lifted in your hands.
• Carry: move around freely while holding it. You can walk, but heavy or stacked carries may slow you (see Configuration).
• Place down: aim at where you want it and use again to set it back, with its full state preserved - chests keep their items, machines keep their progress, mobs are released unharmed.

Picking up a block entity moves the whole tile (and its inventory) rather than just the block, which is what makes it so useful.