Specific movements at advanced levels — lateral work, piaffe, Spanish walk, bow, rear, lie down, and others — absolutely can be taught at liberty, and some practitioners argue that movements learned at liberty are better understood by the horse because the horse must be genuinely responsive to the trainer's cues without any physical backup.
Teaching a specific movement at liberty follows the same operant conditioning principles used for any other horse training — the trainer creates a situation where the desired response is likely to occur, marks and rewards the moment it occurs, and builds the association between the cue and the response through repetition. The tool for marking the desired moment is typically a clicker or word marker that allows precise timing from a distance.
Lateral movements at liberty are often introduced by first establishing them on the ground with light physical contact, then reducing the contact progressively until the horse responds to gesture alone. The critical principle throughout is that at liberty the horse always has the option to refuse, so the motivation must be genuinely positive — the horse must find the interaction rewarding enough to keep engaging.