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Using the moduli space of semiorthogonal decompositions in a smoothprojective family introduced by the second, the third and the fourth author, wediscuss indecomposability results for derived categories in families. Inparticular, we prove that given a smooth projective family of varieties, if thederived category of the general fibre does not admit a semiorthogonaldecomposition, the same happens for every fibre of the family. As aconsequence, 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 seriesis either empty or finite, then any fibre of the family has indecomposablederived category.  Then we apply our results to achieve indecomposability of the derivedcategories of various explicit classes of varieties, as e.g. $n$-fold symmetricproducts of curves (with $0<n<\lfloor (g+3)/2 \rfloor$), Horikawa surfaces, aninteresting class of double covers of the projective plane introduced byCiliberto, and Hilbert schemes of points on these two classes of surfaces.

Indecomposability of derived categories in families