My boyfriend has three little girls, 3, 3 and 5 years old. I try to feed them balanced and correctly portioned meals and snacks even though the 5-year-old is extremely picky. They eat about every 2 hours but always ask for more food every hour. I don't mind giving them more, but when I do they barely eat any of it. If I don't give them more they throw screaming crying tantrums. It's exhausting.
Some kids are "grazers", especially at those ages - they tend to want to eat small portions almost constantly. My youngest (now 6) constantly wanted food until about a year ago, and even within the last year has had flashes of his grazer phase. He used to eat 3 breakfasts, 2 lunches, a mid afternoon snack, and dinner. Although by the time dinner rolled around, he didn't want a big meal. His older sister had a similar phase but wasn't as severe.
We dealt with it by having easy snack foods ready whenever he wanted it (bananas, apple sauce, carrots, cherry tomatoes, cheese sticks). The snacks consisted of easy to eat, easy to serve, healthy foods. And leftovers were always saved and re-served the next time he was hungry.
Issues always arose as we were close to meal times but we managed it as best as possible, often depending how long until dinner was served. Sometimes a very tiny snack was allowed, sometimes nothing.
The concept of 3 main meals a day is very modern and very adult. In my experience, kids are not ready for such a regimented eating schedule, so we found it best to let them eat when they were hungry and let them eventually adapt to eating with the adults when they were ready.
It can get frustrating, I was doing the same thing. I find that when my daughter has a lot of idle time or she's bored she wants to snack and eat constantly all through the day. Just as you say the little ones do at your house. My solution was to keep her active and burning out all that energy. I started walking to the park and after she plays for awhile walking back. Have game time or movie time or tea time and when it's 7pm it's dinner and bath time and she's out for the night. Good luck!
In addition to the excellent answers here, I'd like to add that some kids (adults even) tend to snack when they are bored. So try to keep them active and engaged.
Another is that people tend to snack on habit so keeping them busy will break the habit.
Even that will be exhausting, but hopefully more fun!