So, what exactly did Trump *do* while in DC ?
He isn't the one who brought those documents to FL.
He probably isn't even the one who brought his own underwear to FL from DC, there's no way he was lugging boxes of documents.
Almost certainly he called someone in DC from FL and ordered them to bring the documents. And that was certainly well within his authority as POTUS.
Now, who is responsible for the documents not being removed from FL before Biden was inaugurated ? I can't say, but even if it was explicitly done on Trump's orders, those would have been given in FL.
DC doesn't even enter into it.
They probably forgot that as trump was flying to Florida on air force 1 (before xiden was formally inaugurated) he was still president. He still had a duty to be in charge of nuclear authorization even as he was flying home until the next president took over. There was never a gap of time where "no one" was in charge.