  Factoring ideals in integral domains is a central topic in multiplicative
ideal theory. In the present paper we study monoids of ideals and consider
factorizations of ideals into multiplicatively irreducible ideals. The focus is
on the monoid of nonzero divisorial ideals and on the monoid of $v$-invertible
divisorial ideals in weakly Krull Mori domains. Under suitable algebraic
finiteness conditions we establish arithmetical finiteness results, in
particular for the monotone catenary degree and for the structure of sets of
lengths and of their unions.
