Muscle Coordination is a complicated subject. Indubitably yes, coordination improves with age. Not just with age but also with practice and experience. If this were not so then kids would not learn to crawl, later stand up, later fall and get back up, later take steps, later fall taking steps and get back up, later walk and not fall.
As the brain matures its ability to organize information improves. The older a child becomes then the easier it becomes to process information and translate information from the brain into better or improved motor skills. First attempts at motor skills become successful more often and repeated attempts become lower in number before achieving success.
Muscle memory and brain to muscle communication plays a significant role in learning coordination. When a person is learning a skill for the first time they may have no preconceived ideas about the skill. Or they may have preconceived information in their minds. It could be a visual memory, instructions they have read about, coaching, hearsay, etc. Upon the first attempt the subject gains a mechanical feel for the skill as the muscles and brain work together to create a memory of what the skill feels like. On subsequent attempts the subject is able to refine the muscle groups so that they react more accurately to the demands of the skill. What happens is that the use of muscle groups are brought into better and more targeted use. The larger muscle groups which may have taken too much control at the beginning allow the other muscles to increase their roles. Or the muscle groups that are being used are adjusted with better calibrations. Hence more coordination. If the large muscle groups do all the work or overcompensate then the skill may look clumsy, careless, sloppy, ugly, and repulsive. Its like driving an automobile that without the small steering adjustments needed to compensate for road conditions then the automobile would weave sporadically on the road with large overcompensated adjustments. Yes the automobile can get from point A to point B without the small steering adjustments but if they are not made then the ride may be inefficient or spastic. The muscles and their communication with the brain learn to make those small adjustments to where it soon becomes an unconscious skill. At that point you are free to learn to do other things at the same time that you are driving. You become much more coordinated at driving both physically and mentally. Once the main parts of skills are memorized into the muscles and brain then the gymnast can thereafter consciously refine the skill and use other muscles or the same muscles with better timing to do more things or better things to improve the skill.
A coach will break a complicated skill down into easier parts using drills. Once the gymnast has motor and mentally memorized and refined a part of a skill and reached its level of competency unconsciously, then the gymnast can proceed to concentrate on the next part of the progression without worrying about the previous step. In other words the gymnast is not forced to learn a complicated skill all at once which may be impossible to learn it that way to perfection.