  This is the second of a series of articles devoted to the study of relaxed
highest-weight modules over affine vertex algebras and W-algebras. The first
studied the simple "rank-$1$" affine vertex superalgebras
$L_k(\mathfrak{sl}_2)$ and $L_k(\mathfrak{osp}(1\vert2))$, with the main
results including the first complete proofs of certain conjectured character
formulae (as well as some entirely new ones). Here, we turn to the question of
classifying relaxed highest-weight modules for simple affine vertex algebras of
arbitrary rank. The key point is that this can be reduced to the classification
of highest-weight modules by generalising Olivier Mathieu's theory of coherent
families. We formulate this algorithmically and illustrate its practical
implementation with several detailed examples. We also show how to use coherent
family technology to establish the non-semisimplicity of category $\mathscr{O}$
in one of these examples.
