In this paper:

*R. A. Brualdi, Permanent of the direct product of matrices, Pacific J. Math. 16 (1966), 471482*

(the Kronecker product is called here 'direct product') it is shown that, if $A$, $B$ are nonnegative matrices with order $m$, $n$, respectively, it holds true that:
$$ \operatorname{per}(A \otimes B) \geq \operatorname{per}(A)^n \operatorname{per}(B)^m $$
where the equality holds iff $A$ or $B$ has at most one nonzero term in its permanent expression. Moreover, there exists a minimal number, denoted in the paper by $K_{m,n}$, such that:
$$ \operatorname{per}(A \otimes B) \leq K_{m,n} \operatorname{per}(A)^n \operatorname{per}(B)^m $$
These numbers satisfy the inequality:
$$ K_{m,n} \geq \frac{(mn)!}{(m!)^n (n!)^m} $$
and it is conjectured that we actually have an equality.

Furthermore, in the paper:

*Marvin Marcus, Permanents ot direct products, Proc. Amer. Math. Soc. 17:226-231
(1966)*

it is proven that if $A$, $B$ are positive semidefinite Hermitian square matrices of order $m$, $n$, respectively, then:
$$ \operatorname{per}(A \otimes B) \geq \left (\frac{1}{n!} \right )^m \left (\frac{1}{m!} \right)^n \operatorname{per}(A)^n \operatorname{per}(B)^m $$

Equality holds iff at least one of $A$, $B$ has a zero row.