% ---------------------------------------------------------------- The Beurling\,--\,Gelfand spectral radius formula for the Fourier transform on $\mathbb{R}$ (say) states that \linebreak $\lim_{n\to\infty}\lVert f^n\rVert^{1/n}_1= \sup_\chi\big\lvert\widehat{f}(\chi)\big\rvert$ for any $f\in L^1(\mathbb{R})$, where $f^n:=f\ast\cdots\ast f$ denotes $n$-fold convolution of $f$ with itself, $\widehat{f}$ is the Fourier transform of $f$, and the supremum extends over all characters (i.e., the exponentials $e^{2\pi i\xi}$) of the group $\mathbb{R}$. This extends to measures, relating $\lim_{n\to\infty}\lVert\mu^n\rVert^{1/n}$ to the Gelfand transform $\widehat{\mu}$ (which coincides with the Fourier transform of $\mu$ when restricted to characters), where now $\lVert\ \rVert$ is the total variation norm. We establish such a formula for measures in compact groups, and more generally semi-direct product of the form $A\times_\varphi K$ where $A$ is Abelian and $K$ is compact; these include all Abelian groups, all compact groups, and also the Euclidean motion groups (whence the name locally compact motion groups). We then apply this formula to obtain some known results about norm convergence of random walks on compact groups to the uniform distribution, and some new results about random walks on general motion groups; in particular we characterize mixing and ergodicity of random walks on such groups by means of their Fourier transform.