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If we get a morphism $f : X \to Y$ of schemes over $k$, how should I define the direct image functor $$ f_* : Crys(X) \to Crys(Y)? $$ By a crystal I mean a quasi-coherent sheaf $M$ on $X$ with a collection of isomorphisms (a stratification) between the pullbacks $x^*M \simeq y^*M$ whenever $x$ and $y$ are morphisms from $Spec(R)$ for a commuting ring $R$ which give the same morphisms from $Spec(R/Nil(R))$. Also there are other conditions when $f : Spec(R) \to Spec(S)$ but I omit these.

So my question is how to define the supposed direct image functor, and any help with this would be very appreciated.

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If $f$ is quasi-finite, you can follow chapter IV of Berthelot's thesis (Springer Lecture notes 407). Otherwise, you will need to enter the derived world, much like in the case of D-modules. Gaitsgory-Rozenblyum defines a crystalline pushforward by taking a derived pushforward of ind-coherent sheaves along the corresponding morphism of de Rham stacks.

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