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.