  The property of admitting an astheno-K\"ahler metric is not stable under the
action of small deformations of the complex structure of a compact complex
manifold. In this paper, we prove necessary cohomological conditions for the
existence of curves of astheno-K\"ahler metrics along curves of deformations
starting from an initial compact complex manifold endowed with an
astheno-K\"ahler metric. Furthermore, we apply our results providing
obstructions to the existence of curves of astheno-K\"ahler metrics on two
different families of real $8$-dimensional nilmanifolds endowed with invariant
nilpotent complex structures.
