Of course you can't really "deposit" foreign coins into a US dollar denominated bank account without converting the currencies, but you have to understand how most sorters work. Most sorters have a magnet by the place where the coin goes down then giant wheel that spins around with a cover on it that restricts what coins can go through. An older Canadian cent that is non magnetic is the same diameter as a US cent so of course it will go through. Newer copper plated steel Canadian cents are magnetic and are usually rejected, as are non-silver Canadian dimes, quarters (and half dollars). Different coin sorters use different mechanisms to sort however, some like Coin Stars are good at rejecting coins of the wrong diameter and weight so silver coins will not go through nor will most foreign coins. Others like coins from Panama are made at the US mint and other than the design are indistinguishable from US coinage of that period because they use the same blanks.
In general, the most common (or likely) foreign coins to encounter while CRHing are:
Pennies: Canadian cents, Bahamas cents, and possibly 2 cent Euros
Nickels: Canadian nickels, many different coins/tokens
Dimes: Canadian dimes, British/Commonwealth threepence, British/Commonwealth sixpence, British 5p (new style)
Quarters: Canadian Quarters, South Korean 100 Won, British/Commonwealth shillings, British 5p (old style), British 10p (new style)
Half-Dollars: British/Commonwealth (pre-Decimal) half crowns, British/Commonwealth florins, Mexican 10 Peso (especially in bags, less so for boxes), British 10p (old style), Indian Rupee, Panama half-Balboa, various tokens