Any age that could coherently express an intent in words: "I want you to be my mommy instead".

For babies, this would work like expedited parent-initiated adoption. Probably anyone still holding the baby a month after the birth is determined to make a go of it.

After children start to socialize they would be exposed to this as a pervasive cultural thing among their peers. "Yeah, I traded up to Miss Smith, she's really nice", "Bobby got given away, and he doesn't like them, so he's been asking around, but I think he'll have trouble because everyone knows he's rude".

You were raised in a culture where you get an allocation of at most two parents and that's it. Instinct insists that if they die you starve - naturally the thought of abandonment panics you. I think what this system puts in place instead is more like a tribe where the aunts and uncles chip in with the parenting and the child runs to whoever is closest when they want a hug. Leaving a parent might be a terrible wrench, but the feeling would be there that "I could go back, I could have both at once if they agree, nothing's final".

But of course I'd have to see it run.