  Using the moduli space of semiorthogonal decompositions in a smooth
projective family introduced by the second, the third and the fourth author, we
discuss indecomposability results for derived categories in families. In
particular, we prove that given a smooth projective family of varieties, if the
derived category of the general fibre does not admit a semiorthogonal
decomposition, the same happens for every fibre of the family. As a
consequence, we deduce that in a smooth family of complex projective varieties,
if there exists a fibre such that the base locus of its canonical linear series
is either empty or finite, then any fibre of the family has indecomposable
derived category.
  Then we apply our results to achieve indecomposability of the derived
categories of various explicit classes of varieties, as e.g. $n$-fold symmetric
products of curves (with $0<n<\lfloor (g+3)/2 \rfloor$), Horikawa surfaces, an
interesting class of double covers of the projective plane introduced by
Ciliberto, and Hilbert schemes of points on these two classes of surfaces.
