I usually get a few pings from multiple villagers about public works projects a day. Or villagers just wanting to buy items in my inventory. It's pretty random and fluent.
Generally, you can get multiple pings per day. Just stay out of your villagers' sight for a few minutes (about 3), then walk in front of them, and one of them will ping you about anything, possibly even just random conversation about them finding fruit on the ground.
You can only get one ping to move out per day, at most. You might not even get one. If your villagers talk about rumors and how they spread (in normal conversation), no one will move out. A villager can also move out without pinging you.
With PWPs, you can get multiple requests per day, but it can be really hard, and probably gets rarer with each one (and it's already pretty hard to get one, sometimes).