N-fiber-full modules

Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study $N$-fiber-full up to $h$ $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if $mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $ile h-1$. And we show some applications of this result extending the recent result on squarefree Grobner degenerations by Conca and Varbaro.