How do air-breathing fish swallow without flooding their gut?
A dolphin isn’t a fish, it’s a mammal. Air-breathing fish don’t exist but it’s actually not that hard to swallow an object while still keeping some liquid in your mouth if you put it to the right place in your mouth first. It’s not like trying to eat a bread roll because that would fall apart in water while fish obviously don’t. Some species actually hit the fish out of the water then go up and catch them in their mouths but that’s quite rare. Whales, on the other hand, are filter feeders so use the kind of system you described to seive the food and expel the unwanted water.