https://en.wikipedia.org/wiki/Exigent_circumstance
"An exigent circumstance, in the criminal procedure law of the United States, allows law enforcement, under certain circumstances, to enter a structure without a search warrant or, if they have a "knock and announce" warrant, without knocking and waiting for refusal."
Further, does the law, as written, allow detainment of individuals while those reasons are decided?
Until someone can describe the legal link between the fact it's a missing child and the conclusion these are exigent circumstances from a legal perspective, I can only interpret the pointing out that it's a missing child as a plea to emotion at best and a thought terminating cliche at worst.