We study the asymptotic behaviour of Gagliardo $H^s$ seminorms on thin films of thickness  tending to 0. In the light of the results by Bourgain, Brezis and Mironescu (BBM) and Maz'ya and Shaposhnikova (MS), we compute the critical scalings of epsilon for which the asymptotic behaviour as $s\to 1$ and $s\to 0$, respectively, can be described by a dimensionally reduced functional. In the case $s\to 1$ the scaling highlights a combined effect of the geometrical dimensions and the relevant range of interactions in the Gagliardo seminorms, and the limit is a $H^1$ seminorm. The asymptotic behaviour highlights a separation of scales and a compatibility with the BBM result. In the case $s\to 0$ the scaling is purely geometrical and leads to a $H^{1/2}$ seminorm, differently from the MS result. The analysis relies on two different approaches to compactness, and is extended also to $s$ converging to a limit in $(0,1)$.