They need a consequence, not only for the balls/toys, but for the trash, and for the disturbances. Adopt your no-nonsense teacher voice. And so...
"Guys, this is happening too much. It's annoying, and when it happens this much, it's rude that you're not being more careful to keep your stuff out of our yard. So here are the new rules. When something comes into our yard, you get to come over here and ring the doorbell. ONCE. Got that? ONE TIME. Then you WAIT for someone to get there. I'm not Superman, and I can't fly to the door in one second. You wait politely and quietly for someone to come, and you don't look in the house or bang on the door and lay on the doorbell so it keeps ringing. That is rude. Got it?" (wait for them to acknowledge)
"OK, after you tell us about it, we'll go to the back yard and check out what's there, and you can go home." (cue the curious looks from the kids). Because see, we're done getting your stuff whenever you want it. This has been happening several times a night, and we're done. We're not dogs, and we don't play fetch. That (point) is your yard, and this (point) is OUR yard. When it happens that much, I can tell you're not even trying to keep your stuff in your own yard. If you don't care enough to keep it in your own yard and play carefully with it, then I don't care if you have it back right away. (shrug like it's a no brainer) I'll go to the back and get it, but for every single thing that I find in my yard, I'm keeping your toys for one day. So if I find two toys, I keep them for two days. If I find a ball and ten candy wrappers, then guess what? That's eleven things, and I'm keeping all of the stuff for eleven days. Yeah, I know you throw your garbage in our yard. (pause) That's rude too, and that's got to stop. And you know what else? I'm also going to add a day for every time you bug us about it. You tell us ONCE. And you totally control how quickly you get your toys back by how well you keep stuff in your own yard. Do you guys understand?"
Now, I know that I could totally pull off this attitude/voice, but it might be an acquired talent, so you may need to practice.
But if I'm getting this right, you'd like everything to stop coming into your yard. So therefore, there needs to be a hard consequence for every single thing that they throw there. If the parents get mad, so what? Their kids are using your yard as a trash bucket and there's no way that's acceptable. Make it hurt for them for a bit.
It may mean a bit of bookkeeping on your end, and you should probably save the wrappers as well as the toys as proof of your 'timeframe', but it could work. And it just could teach them some patience and awareness, b/c they'll be acutely aware of how much they throw in your yard if they know that every little thing means a day without a toy. And the beauty of it is that if they even ask you how much time they have left, that's another day b/c they bugged you. You have to make them own their actions. Young kids, yes. But they can not throw the stuff as easily as they can throw it.