Supposedly (consciouss) movement is started in the cerebral cortex. But some time ago I've read a research which stated it might be possible some movements (not specified which ones, probably mean automated ones) start in the brain stem. That research was in one of those research file websites, it wasnt an article and it didnt have any mainstream visibility. Then I didnt hear anything more in this line, but you hear many times about these babies "born without a brain" who lacks most of its brain, 80% of its brain or who lacks cerebral cortex yet they move. Where are those movements started? Can motor neurons in the brain stem start movements?
Yes, movements (including fairly complex ones) can start in or be driven by the brainstem (and spinal cord).
Central pattern generators are key to producing coordinated rhythmic movements, including those for locomotion, chewing, and breathing: these are also located in the brainstem and spinal cord.
There have been many experiments done to study brainstem control of locomotion (see Shik and Orlovsky, 1976), particularly in decerebrate cats. You can induce a walking motion independent of cerebral control in these animals just by moving the floor (e.g., using a treadmill).
More generally, voluntary motor commands from cortex are mostly just a "suggested plan" - the specific patterns of muscle activation are determined in the brainstem and spinal cord, with input from proprioception and modulation by the cerebellum and brainstem nuclei such as the red nucleus.
In humans, spontaneous movements are also common in ~50% of brain-dead patients (Saposnik et al. 2005), although definitions of brain death typically include the brainstem, so these should be thought of as spinal rather than brainstem movements.
Saposnik, G., Maurino, J., Saizar, R., & Bueri, J. A. (2005). Spontaneous and reflex movements in 107 patients with brain death. The American journal of medicine, 118(3), 311-314.
Shik, M. L., & Orlovsky, G. N. (1976). Neurophysiology of locomotor automatism. Physiological reviews, 56(3), 465-501.