  We extend recent results on affine Volterra processes to the inhomogeneous
case. This includes moment bounds of solutions of Volterra equations driven by
a Brownian motion with an inhomogeneous kernel $K(t,s)$ and inhomogeneous drift
and diffusion coefficients $b(s,X_s)$ and $\sigma(s,X_s)$. In the case of
affine $b$ and $\sigma \sigma^T$ we show how the conditional Fourier-Laplace
functional can be represented by a solution of an inhomogeneous
Riccati-Volterra integral equation. For a kernel of convolution type
$K(t,s)=\overline{K}(t-s)$ we establish existence of a solution to the
stochastic inhomogeneous Volterra equation. If in addition $b$ and $\sigma
\sigma^T$ are affine, we prove that the conditional Fourier-Laplace functional
is exponential-affine in the past path. Finally, we apply these results to an
inhomogeneous extension of the rough Heston model used in mathematical finance.
