If you are just cutting a large pile of small pieces on a large board, and then trying to get it into a small container, that is going to be messy indeed. The first difficult thing is to change how you think of it. Instead of merrily chopping everything, then going on to the next step, you will need some planning. And in many cases, this will involve chopping smaller amounts at once.
The best way is to always choose a board whose width is smaller than the container's opening. Restaurant cooks may chop on huge boards, but they also have huge pots waiting on the burners. If you are cooking for one and using a 20 cm wide pot, then ideally you will be chopping on a 18 cm wide board, even if those are being nominally sold as "breakfast board" and not "cutting board". And I know that you said in a different comment that you personally will not replace your own boards, but it may indeed turn out that you bought the wrong boards for your own cooking needs.
Chopping on a small board may lead to a situation where you cannot easily chop all of your produce at once, because it starts falling off the board. In that case, you should simply chop in several steps, removing chopped food from the board in regular intervals. For example, chop two carrots at once, remove, then two more carrots, etc. In some cases, you will get away with emptying everything into the cooking vessel, e.g. for slow cooking. In other cases (like onions waiting to be sauteed) this will not work, because the whole batch has to hit the vessel at once. For that, you should do better mise en place, first placing the chopped food into bowls, and only after you are ready, starting cooking with the prepped food.
If you absolutely insist on using a large board for food chopped into small pieces, then mise en place and/or chopping in portions is also the way to go. When you chop in portions, you first order scrape food into a long and thin pile with your knife - for a carrot, you don't even need to do it, but for something like a cubed bell pepper, try preshaping it like a row of carrot discs - and then you need a low angle and precise movements to guide it into the pot without spilling on the outside. If it is rather long (like a leek or a large carrot), don't try to scrape it from the "tail", instead scrape the "head" first, then work your way up the board. This is the same principle as in making spaetzle, only the food is not a single mass.
If you give yourself time to learn, you can also use the diagonal method. First, you shape your long pile. Then you hold the board over your bowl with the left hand, such that the pile's long side is roughly parallel to the counter, slightly leaned towards the pot on one end. With your right hand, you place your knife parallel to the pile. Now you start rotating the board relative to the knife's blade, such that the first few pieces closest to the knife's blade butt fall into the bowl. At the same time, you move the board to the right, such that the next pieces of the pile arrive at the knife's butt. The motion is difficult to explain in words, but it is not that different from honing a knife - only that you have a whole cutting board in place of the honing steel, and your knife blade stays vertically to the plane of the board, instead of almost in the same plane.
When you use mise en place, you can alternatively chop everything into a large conical pile, then grab most of the pile with your hands or large spoon and throw into the bowl, then use the long-pile method on the remaining pieces. Orienting the long pile towards a corner and then pointing the corner into the pot or bowl also helps.
For some very small and/or sticky things like chopped herbs, you can also use a sweeping method - leave the board laying down, hold something shovel-like with an edge on the board (ideal would be one of those small shovels for scooping nuts, but even something like a metal spatula can work) and use your knife to brush the herbs into the improvised "shovel". Then place it from the shovel into the mise en place bowl, or use the shovel itself for mise en place, if it is something fillable.