  We focus on the positive dimensional fibres of the Prym map
$\mathcal{P}_{g,r}$. We present a direct procedure to investigate infinitely
many examples of positive dimensional fibres. Such procedure uses families of
Galois coverings of the line admitting a 2-sheeted Galois intermediate
quotient. Then we generalize to families of Galois coverings of the line
admitting a Galois intermediate quotient of higher degree and we show that the
higher degree analogue of the aforementioned procedure gives all the known
counterexamples to a conjecture by Xiao on the relative irregularity of a
fibration.
