As team physician for the Norfolk Tides, I do exit physicals at the end of every season. Some of the Tides players move up to the Orioles as a "September call up". This year Josh Rogers, Luis Ortiz, Chance Sisco and Steve Wilkerson moved up to Baltimore and you can catch them on TV, but for most players it marks the end of their season.
During exit exams, I always ask the pitchers what they plan to do at the end of the seaon, and for almost all of them the answer is the same, "Nothing!". They shut it down, no throwing for about eight weeks. During that time they may work on general conditioning and strength training but they give their arms a period of rest. After that rest period, they gradually resume their throwing routine. Some players will play winter ball, but this is usually younger guys who did not get enough innings during the season, or who missed a period of the season due to injury. Basically, the guys who need to work their arms some more.
Contrast this to most high level youth baseball players. They play all year round, including winter workouts. Unfortunately, kids need that rest even more than the pros do. Their muscles and bones aren't fully developed and they are not equipped to take the stress of year round play. This is what leads to injuries. Furthermore, year round play does not help performance. By the end of the season, many of the professional pitchers I see have significant fatigue which leads to rotator cuff weakness and shoulder stiffness. Many guys start to notice a dip in their velocity by the end of the year. They need that period of rest and stretching to get their arms ready for the next season.
The bottom line is: Make sure your little leaguer is getting at least two months of "no throwing" at some point during the year. They need to rest to prevent injury and improve performance. During this time period they should not be playing other over head sports such as throwing a football, swimming or volleyball.