  We study the distribution of families of multiplicative functions among the
coprime residue classes to moduli varying uniformly in a wide range, obtaining
analogues of the Siegel--Walfisz Theorem for large classes of multiplicative
functions. We extend a criterion of Narkiewicz for families of multiplicative
functions that can be controlled by values of polynomials at the first few
prime powers, and establish results that are completely uniform in the modulus
as well as optimal in most parameters and hypotheses. This also significantly
generalizes and improves upon previous work done for a single such function in
specialized settings. Our results have applications for most interesting
multiplicative functions, such as the Euler totient function $\phi(n)$, the
sum-of-divisors function $\sigma(n)$, the coefficients of the Eisenstein
series, etc., and families of these functions. For instance, an application of
our results shows that for any fixed $\epsilon>0$, the functions $\phi(n)$ and
$\sigma(n)$ are jointly asymptotically equidistributed among the reduced
residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\log
x)^{(1-\epsilon)\alpha(q)}$, where $\alpha(q) = \prod_{\ell \mid q}
(\ell-3)/(\ell-1)$; furthermore, the coprimality restriction is necessary and
the range of $q$ is essentially optimal. One of the primary themes behind our
arguments is the quantitative detection of a certain mixing (or ergodicity)
phenomenon in multiplicative groups via methods belonging to the `anatomy of
integers', but we also rely heavily on more pure analytic arguments (such as a
suitable modification of the Landau-Selberg-Delange method), -- whilst using
several tools from arithmetic and algebraic geometry, and from linear algebra
over rings as well.
