In Elio's case, he would do his best to calm them down first; he would bring out Jupiter or Estio since the two of them are very friendly toward strangers - especially if they're children. The two would try to get the lost child's attention by performing tricks as if they were in the Performance Stage of a Contest, which would normally be successful due to a young child's natural sense of wonder. Then, Elio would piggyback them to the nearest bathroom to get them cleaned up, and probably buy them a treat (ex: food, a balloon, a toy, etc.) to make sure they remain calm; he would even allow Jupiter or Estio to continue playing with them if it'd make the lost child feel better. Only then would he tenderly ask them what's wrong and do his best to assist them out in any way he could: from tracing the child's steps to using his other Pokemon to a wide search of the nearby area. If necessary, he would get Officer Jenny involved.

This would be one of the few times where Elio's "big brother vibes" would naturally come out in full since he's used to caring for children and/or his juniors - making it easier for him to gain their trust.