I can't speak to crossbows, but with regards to muskets and early rifles, soldiers and hunters absolutely did do this, but only in special circumstances (including, rarely, defending a position), because it's very difficult to haul around multiple guns and shot and powder for these when you need to march, and it's rather pointless once you've fired all the extra guns. Then, you're left with three, say, empty muskets, and it doesn't make any sense to shoot all three, then reload all three, then shoot all three again, etc. It would defeat the purpose. Eventually, you're just going to get back to reloading and firing the one musket, but now you've got to haul around two extra useless sticks.

Also, it's important to remember that fighting units of the muzzle-loading days were just that: units. They were effective firing en masse and were thought of, in a sense, as a large block of musket fire, rather than a bunch of individual soldiers. That means that everyone needs to fire at the same time, or at least each rank needs to fire at the same time if volley fire is used, in order for the unit to be effective. This also means that officers were trained to use their units deliberately and effectively -- which also entailed slowness of targeting. Quickly switching weapons to fire another salvo was not at a premium.

Now, this gets into the strategy and general concepts of warfare that were contemplated by the great military minds and individual military commanders at the time. Thing is, the concept of firepower -- within which rate of fire is a small part -- is a relatively modern one, and was not properly or fully understood by armies at this time, or even much later times. For example, by the time of the Civil War, repeating, lever-action rifles were available, but were rejected by the Union army quartermaster general in favor of the old, muzzle loading rifles, because they were smaller caliber and no clear advantage could be derived from faster firing with Napoleonic-style tactics. Later in the Civil War, individual units did adopt these weapons, but this was ad hoc.

So, while an individual, or small group might use the switching-off tactic (especially if some individuals were good shots while others were fast loaders), for the most part, in open warfare, this was not an effective way to put steel on target.

As luck would have it, here is a nice (if fictional) depiction of both of these things in action -- individuals switching off weapons, and a unit of musketmen working in tandem under the guidance of an officer.