I know it is a complaint issue in several worlds. For a rookie or novice owner, I can understand how it happens. Sometimes they ignore the problem and hopes it fixes itself, in most cases it won't without intervention. Maybe they don't know the best way to fix the problem and don't ask for help. I also know that not every owner can check their teams after every playing cycle which can cause unexpected problems. Real world stuff does happen, but the commish should be notified if at all possible.
As some have said, turning the AI for the minors can help. The only warning I have for that is if you handle draft signing like I do. If you haven't signed everyone and you have the money, the AI will do it for you.
Most of this is about pitching fatigue, but don't forget about position fatigue also, a player is more apt to be injured the lower he goes. I used to complain about owners playing players out of position in the minors, not any longer. I would rather see a fresh body at a position whether he can play it or not than have a good player on the DL.
There are a couple unavoidable situations I have been in where there is no immediate fix. Starting pitchers going one or two innings before being injured, not a problem if it happens once, but I have had that scenario happen on back to back days. That is a tough recovery when you already have a roughed up pen. I had a 21 inning Rookie league game in another world this season, try fixing that in a day or two. But still, fixing the problem only takes a day or two if you are proactive.
In the minors I don't worry about my pitching that much. What I do care about is quantity and quality. In Low A, I normally carry 17 pitchers and I try to carry 14 or more at all other levels. The 17 is on purpose, it will allow me to control problems that may arise at High A and the Rookie League. A spare or two at AA and AAA will allow me to control problems there.
Most generally it is stated never to demote a player ever, I agree with that under most circumstances. If the player is going to be a ML quality player one day, never demote him. However, if the player has no relevance to the ML team now or in the future and he is needed at another level, it sure isn't going to hurt him or your organization all that much. He may opt to leave the following season, but then what difference will that really make?
What I try to do on the make up of a team is to have 10 starters, their makeup usually consists of good stamina, control and pitches, notice I don't care about splits. From those 10 I choose the best 5 as starters with ML prospects getting first dibs of course. Then I might put in three relief pitchers to round out the pen. They are usually closer types, but I seldom run a closer in the minors. In reserve I just care about having some extras. At the beginning of the season I call all of them Setup A so I don't have to figure out what they are. About 30 games into the season, I determine if they are Setup A or B depending on their ERA, the worst becomes B's. If I have a pen starting pitcher performing better than a starter, I won't hesitate to change them. I never run with a mop up pitcher either, I find that roll rather useless. I also try to make sure pen pitchers are limited a little so they don't get over pitched and fatigue below a days rest.
If I run into trouble at another level, I will promote a guy up the line to fix the problem if necessary. I start the movement from Low A of course. Then I go and find another body to replace him from IFA, FA, WW, tryout camp or Rookie League.
Everyone has a different way of handling the draft and moving players once the season is over. But here are a few things that could be helpful. After rollover I try to have my Rookie League already stocked with 20 players. I get them from tryout camp and the WW or left overs from the previous season. It also provides for a handy pool of extra players for the first third of the season. Most generally I try to find starting pitching types or middle relievers.
Why do it this way? I am not burdened with needing to find a certain type of player or needing to sign a player I draft immediately. What I normally do in the draft is find the best player for the first round and a needed player the second round. I try not to draft relief pitchers at all unless they are first round quality. I set the draft up to draft SP only in the pitching realm as described earlier though I pay more attention to splits. Normally I only sign my top ten picks and the best players that are better than what I already have on the roster. The top 3 may wind up in Low A. I used to be against this but it does have some advantages. Like resting position players with like abilities instead of a lessor quality player. In all reality, I have not seen any player actually advancing at a faster or better pace. Also, if I need another player, I can always sign him off the draft list.
At the end of the season, I promote the best players out of the Rookie League, release the real bad ones, usually ones that have absolutely no value or have a bad health. Then I sign the rest of the draft picks to repopulate and watch the WW also. I might have players that are career rookie league players, if they retire the next season, it won't hurt that much. Actually, I don't mind career minor league players retiring, it is replacing them that is a pain sometimes.