GAGA problems for the Brauer group via derived geometry

This paper is dedicated to a further study of derived Azumaya algebras. The first result we obtain is a Beauville-Laszlo-style property for such objects (considered up to Morita equivalence), which is consequence of a more general Beauville-Laszlo kind of statement for quasi-coherent sheaves of categories. Next, we prove that given any (derived) scheme $X$, proper over the spectrum of a quasi-excellent Henselian ring, the derived Brauer group of $X$ injects into the one of the Henselization of $X$ along the base, generalizing a classical result of Grothendieck and a more recent theorem of Geisser-Morin. As a separate application, we deduce that Grothendieck's existence theorem holds for the stable $\infty$-categories of twisted sheaves even when the corresponding $\bbG_m$-gerbe does not satisfy the resolution property, offering an improvement of a result of Alper, Rydh and Hall.