Roots of line bundles in a family

Question

Let $ \pi : X \rightarrow B $ be a family of compact complex manifolds parametrized by a connected base $ B $. (By this I mean $ \pi $ is a proper holomorphic submersion.) Let $ L $ be a holomorphic line bundle on $ X $ and $ n $ a positive integer. What can be said about the locus of points $ b \in B $ such that $ L|_{X_b} $ on $ X_b $ has an $n$-th root?