NARA is managing the files and always has been. I just showed NARA and the US military moved the documents in 2016.

NARA manages presidential libraries. There is some confusion because a presidential library has two meanings, the collection of documents managed by NARA, and the museum that is built with private money. Even when documents are displayed in a presidential museum, NARA is in charge of those.

There's no building for a Trump Presidential library/museum and as far as I know, no plans as of yet. But NARA still runs the library. see here https://www.trumplibrary.gov/

NARA has been archiving the Obama documents since 2016 (some were delivered early) and they are now available for FOIA requests which have been going on since January 2022....the PRA states the records become available 5 years after the president leaves office.

NARA leasing existing buildings has been going on for decades. GSA finds a big space and NARA takes it over and secures it and preps it for archving.

After Bush I, NARA leased an old bowling alley in Texas and made into a NARA warehouse before the Bush Library and Museum was built https://www.latimes.com/archives/la-xpm-1994-06-26-mn-8638-story.html

And they secured it

Uniformed guards patrol the premises. There are closed-circuit television monitors and sophisticated electronic detectors along walls and doors. Some printed material is classified and will remain so for years; it is open only to those with top-secret clearances.

NARA took over and old car dealership in Little Rock for Clinton. For Bush they took over a warehouse that today is a factory. https://www.nwaonline.com/news/2009/dec/21/bush-library-digs-legacy-20091221/

Classified documents are stored in a sealed area of the warehouse.