Possibly. Parson divided them by move. Higher move may be due to a higher level or due to different types.
Yeah, I would have assumed that was due to breed. Leveling, to me, seems more likely to bring about a boost to other stats in a heavy like dwagons, Although I suppose it could vary by breed, so Pink Dwagons are better skirmishers, Blue are Heavy hitters, and Red is all-around, which would change what happened when they leveled.