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Let Fq[t] be the polynomial ring over the finite field Fq. For arithmetic functions ψ1, ψ2 : Fq[t] → C, we establish that if a Bombieri-Vinogradov type equidistribution result holds for ψ1 and ψ2, then it also holds for their Dirichlet convolution ψ1 * ψ2. As an application of this, we resolve a version of the Titchmarsh divisor problem in Fq [t]. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in Fq[t].

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