  Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived word. We show
that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are
non-commutator words in disjoint sets of variables, thus proving a generalized
version of a conjecture of Azevedo and Shumyatsky. This applies in particular
to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are
non-zero integers. Our approach is via the study of values of $w$ on normal
subgroups, and in this setting we obtain the following result: if
$N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values
$w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup
generated by these values, i.e. $w(N_1,\ldots,N_r)$, is finite.
