Can 'along' be omitted without changing the meaning?It means (move) forward, or (move) in that direction. It’s a very common phrasing and I understand the whole thing as “keep moving”. Don’t stop here.
If they want to start moving now not that they were moving, can't 'along' be used?"Move" by itself means "change to a different location", for example, if where someone is creates an inconvenience for someone else.
"Move along" means "continue moving" / "don't stop", probably in an effort to keep traffic moving.