  We prove that, for arbitrary Dirichlet $L$-functions
$L(s;\chi_1),\ldots,L(s;\chi_n)$ (including the case when $\chi_j$ is
equivalent to $\chi_l$ for $j\ne k$), suitable shifts of type
$L(s+i\alpha_jt^{a_j}\log^{b_j}t;\chi_j)$ can simultaneously approximate any
given analytic functions on a simply connected compact subset of the right open
half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and
satisfy certain conditions. Moreover, we consider a discrete analogue of this
problem where $t$ runs over the set of positive integers.
