How do "on hit" effects work with channeled / DoT skills?

It depends on the skill in question.

First off, channeled skills are exactly the same as normal skills in terms of triggering things, except you do not need to re-cast the ability between each tick. Each channeled skill has a "Tick" (that occurs more often the faster attack speed you have), and it is these ticks that trigger on-hit effects. Each "Tick" is when you pay the resource cost of the skill.

Now, whether channeled skills are better depends on the on-hit effect you're trying to use. Anything with a listed % chance on hit, e.g. 1.1% chance to fear on hit, will be better the more attacks/ticks you make. Something listed as a "chance to happen", such as the wizard's "Critical Mass" i. e., "Critical Hits have a chance to reduce the cooldown of your spells by 1 second. " are most likely set up for ppm (proc per minute), which means you have a better chance of triggering them with a slow weapon, with the intent that you get the same number of triggerings, procs, over any given time period as a faster weapon.

Lastly, bear in mind that all that matters is the "Tick" times. If Rapid Fire shoots 10 arrows per Tick, all 10 of those arrows represent a single chance on hit (much like the Monk's Way of a Hundred Fists)

So to conclude, no, it is not better to use lots of channeled / DoT skills, for the simple reason that they don't provide any innate advantage. Unless you're fishing for Procs, you'll gain a much better chance to trigger things by using a faster weapon than any particular ability.