Given a positive continuous function
$\mu $ on the interval
$0\,<\,t\,\le \,1$, we consider the space of so-called
$\mu $-Bloch functions on the unit ball. If
$\mu \left( t \right)\,=\,t$, these are the classical Bloch functions. For
$\mu $, we define a metric
$F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of
$\mu $-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.