I would check two things:
1. Front derailleur height
2. Front dérailleur cage "yaw"? The angle of the outside cage relative to the chainring.
3. Front chute area might be damaged (where chain passes through at front of front derailleur)
4. Guard when shifting into the small ring.
You want the front bit of the front derailleur to guide the chain onto the chainring. The rest of it is there to guide the chain into that front chute.
1. Most front derailleurs I see are higher than they need to be. I keep mine as low as possible. This prevents the chain from slipping between the cage and the chainring and derail. Picture of my front derailleur height:
2. In addition to the "too high" front derailleur I also see them twisted with the tail toward the centerline of the bike. This is usually due to a combination of the cable tension pulling on the front derailleur and a loose/r clamp. Pulling the cable on a less-than-tightened front derailleur results in the tail moving inward. The chain won't move up to the big ring easily and it'll rub the cage. The natural tendency is to unscrew the limit screw to let the front derailleur move more. This results in a larger cage-chainring gap which encourages the chain to drop.
A secondary result of an inward pointed front derailleur cage is that the front derailleur will let the chain drop when shifting to the small ring.
I don't have a good picture of the tail angle - the only one I could easily find is very distorted due to the lens. Basically I adjust mine so that the outer cage is essentially parallel with the chainring (minus any weird bends in the chute area up front).
3. Front chute damaged or spread. I usually squeeze the front chute area closed just a touch with channel locks or similar. The narrower the chute the more aggressive the front derailleur will shift. Obviously don't damage it. Some older front derailleurs have two very simple flat plates and respond well to this trick. Newer derailleurs tend to be reinforced and shaped much more aggressively and don't respond as well. Of course don't do it on the carbon cages.
4. Dropping the chain to the inside. The front derailleur isn't great for shifting into the small ring. It's impossible to limit the movement effectively with the cage (since it's hard to make a cage work for different size chainrings). Remember also that your chain is under pedaling tension, not like at the rear derailleur, so it takes a lot of force. Combined with less precision it gets tricky.
For me
I use an N-Gear Jumpstop chain guard. I've gotten into the habit of shifting gently but I'll adjust the front derailleur to over shift into the small ring (so it gets down there even when I'm pedaling moderately hard) and use the Jumpstop to catch it.
Hope this helps.