  Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field
$\mathbb{F}_{q}$. For arithmetic functions $\psi_{1}, \psi_{2}:
\mathbb{F}_{q}[t]\rightarrow\mathbb{C}$, we establish that if a
Bombieri-Vinogradov type equidistribution result holds for $\psi_{1}$ and
$\psi_{2}$, then it also holds for their Dirichlet convolution $\psi_{1} \ast
\psi_{2}$. As an application of this, we resolve a version of the Titchmarsh
divisor problem in $\mathbb{F}_{q}[t]$. More precisely, we obtain an asymptotic
for the average behaviour of the divisor function over shifted products of two
primes in $\mathbb{F}_q[t]$.
