In the case of a family of minimal smooth projective varieties *of general type*, we can repeat exactly the same argument as in the case of curves of genus $\geq 2$.

In fact, by a result of Hacon, McKernan and Xu **[HMcKX13]** we know that, if $X$ is any such a variety, we have $$|\mathrm{Aut}(X)| \leq c \cdot \mathrm{vol}(X, \, K_X), $$ where $c$ is a constant that only depends on $d:=\mathrm{dim}(X)$.

Now, it sufficies to observe that the canonical volume $$\mathrm{vol}(X, \, K_X):=\limsup_{m \to + \infty} \frac{d !\; h^0(X, \, mK_X)}{m^d}$$ is deformation invariant, by the celebrated invariance of plurigenera in smooth families proved by Siu **[S98]**.

**References.**

**[HMcKX13]** *Hacon, Christopher D.; McKernan, James; Xu, Chenyang*, **On the birational automorphisms of varieties of general type**, Ann. Math. (2) 177, No. 3, 1077-1111 (2013). ZBL1281.14036.

**[S98]** *Siu, Yum-Tong*, **Invariance of plurigenera**, Invent. Math. 134, No. 3, 661-673 (1998). ZBL0955.32017.