The 13 presidential libraries from Hoover through George W. Bush are federal institutions, not private; so, yes, any incumbent administration has authority over them.
And Obama does not have an official federal presidential library (no FPOTUS after GWB does, or will, under current policy).
And Obama does not have an official federal presidential library (no FPOTUS after GWB does, or will, under current policy).
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The Obama Center is a private institution with no archival component, and no federal presence.
While there are laws and regulations governing the presidential libraries and national monuments, Trump ignores or violates every law, so. 🤷♂️
If he tried that at the LBJ Library, for instance, the University of Texas at Austin would hook 'em.
And during his first term, Trump did delay construction, by withholding the federal permits.