We strongly associate a boxer training in front of a punching bag, and a karateka breaking boards.
Would the boxer or the karateka benefit from cross training, that is, boxer breaking boards, karateka punching punching bag? Is it common to cross train?
Part of the issue is that board breaking is not generally training. It is a form of demonstration. Its practice is basically tied to showing strength and speed of a single strike in a measurable form at a time when we didn't have accelerometers and high-speed cameras. Today, boxers often do something very similar to a board break in purpose where they're shown delivering a punch to a punching bag, and the audience is then shown the measured speed and force of the punch.
What may be a more interesting question is looking at breaking training versus bag work.
Generally, a martial artist engaged in breaking will practice by repeatedly hitting hard surfaces. Masutatsu Oyama, a famous breaker who was known for breaking the horns off bulls, would use trees. In karate, a device called a makiwara is used; this device has found more popular use by practitioners of other martial arts today. In the past, Shaolin and other earlier martial artists would use many different types of devices in order to condition themselves, not always for simply breaking, but using the same concepts used today. For instance, there is Iron Palm, Iron Shin, Iron Shirt, Iron Head, and other types of training which center around conditioning various parts of the body so they could withstand or give blows such as what is seen today in martial arts breaking. Many Chinese systems also are of the school of thought that "internal energy" or Chi is used when breaking, which is not dependent upon muscle strength and body weight.
Namely, part of breaking training involves barehanded, or lightly clad, strikes against unyielding surfaces to make the skin tougher and the bones more dense, and enforcing proper striking technique with barehanded blows to avoid injury. If a boxer plans to fight without gloves, and strike with their full force, then yes, they may benefit from breaking training to build up their endurance.
As regards general bag-work, striking takes energy, obviously, and any fight that doesn't end quickly is going to require you to have a fair amount of endurance. Punching, or kicking, repeatedly at a target like a bag is a good way to build up that endurance. My understanding is that this is the common purpose for boxers striking a heavy bag, that it's not as much about getting their punches stronger, as rather about learning to punch over and over again in quick succession against resistance, so that you can similarly maintain that endurance in the ring. And yes, that is something that karatekas do with their training. Although I'd argue that there's some difference in philosophies between the two styles, boxing focusing on multiple strikes, only sometimes leading to a definite "knockout blow", versus Karate, which tends to be a bit more about delivering that one perfect blow to disable an opponent.
