$(1)$ When $X$ is Calabi-Yau, there is a "derived version" of monodromy around a conifold point. It was also investigated in the framework of string theory, see for instance the paper

Paul S. Aspinwall1 and Michael R. Douglas

"D-brane stability and monodromy"

Journal of High Energy Physics (2002).

$(2)$ When $X$ is a $K3$ surface, it is possible to construct a representation of the group of auto-equivalences $\textrm{Aut } D(X)$ on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on $X$, and to relate this representation to the monodromy of the Hilbert schemes $X^{[n]}$ of points on the surface. See

Eyal Markman

"On the monodromy of moduli spaces of sheaves on K3 surfaces",

J. Algebraic Geom. 17 (2008), 29-99.