Over the complex numbers, a famous theorem of Siu states that the plurigenera $P_m$ of projective manifolds are invariant under deformations. We give examples of families of elliptic surfaces over a DVR of positive or mixed characteristic such that $P_m$ fails to be constant for all sufficiently divisible $m\geq 0$. Time permitting, we will show that (asymptotic) invariance of plurigenera holds for families of quasi-elliptic surfaces.