A seguence of measures $\{\mu_n\}$ on a compact group $G$ is called composition convergent if, for any $i$ , the seguence of measures $\mu_i\mu_{i+1}\dots\mu_{i+n}$ converges as $n\to\infty$ in the ordinary sense to a measure $\nu_i$ . The main property of composition convergent sequences is that a limit of $\nu_i$ as $i\to\infty$ exists and is equal to the uniform measure on a subgroup $g$ , $g\subseteq G$ which we call the base of $\{\mu_n\}$ . We show that for any sequence$\{\mu_n\}$there corresponds a composition convergent sequence$\{\mu'_n\}$obtained from$\{\mu_n\}$by shifts:$\mu'_n=\alpha^{-1}_{n-1}\mu_n\alpha_n$,$\alpha_n\in G$. This fact enables to make use of the above property of composition convergent sequences of measures when dealing with convergence conditions for compositions of different measures$\{\mu_n\}$on$G$.