  Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$
compact. We characterize spread-out probability measures on $G$ that are mixing
by convolutions by means of their Fourier transforms. A key tool is a spectral
radius formula for the Fourier transform of a regular Borel measure on $G$ that
we develop, and which is analogous to the well-known Beurling--Gelfand spectral
radius formula. For spread-out probability measures on $G$, we also
characterize ergodicity by means of the Fourier transform of the measure.
Finally, we show that spread-out probability measures on such groups are mixing
if and only if they are weakly mixing.
