  A \emph{composition} is a sequence of positive integers, called \emph{parts},
having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair
of adjacent parts $x$ and $y$ within a composition such that $x\equiv
y(\text{mod} m)$. Here, we consider the problem of counting the compositions of
size $n$ according to the number of $m$-congruence successions, extending
recent results concerning successions on subsets and permutations. A general
formula is obtained, which reduces in the limiting case to the known generating
function formula for the number of Carlitz compositions. Special attention is
paid to the case $m=2$, where further enumerative results may be obtained by
means of combinatorial arguments. Finally, an asymptotic estimate is provided
for the number of compositions of size $n$ having no $m$-congruence
successions.
