  In this paper, we develop a comprehensive weighted theory for a class of
Banach-valued multilinear bounded oscillation operators on measure spaces,
which merges multilinear Calder\'{o}n-Zygmund operators with a quantity of
operators beyond the multilinear Calder\'{o}n-Zygmund theory. We prove that
such multilinear operators and corresponding commutators are locally pointwise
dominated by two sparse dyadic operators, respectively. We also establish three
kinds of typical estimates: local exponential decay estimates, mixed weak type
estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de
Francia extrapolation for abstract multilinear compact operators, we obtain
weighted compactness for commutators of specific multilinear operators on
spaces of homogeneous type. A compact extrapolation allows us to get full range
of exponents, while weighted interpolation for multilinear compact operators is
crucial to the compact extrapolation. These are due to a weighted
Fr\'{e}chet-Kolmogorov theorem in the quasi-Banach range, which gives a
characterization of relative compactness of subsets in weighted Lebesgue
spaces. As applications, we illustrate multilinear bounded oscillation
operators with examples including multilinear Hardy-Littlewood maximal
operators on measure spaces, multilinear $\omega$-Calder\'{o}n-Zygmund
operators on spaces of homogeneous type, multilinear Littlewood-Paley square
operators, multilinear Fourier integral operators, higher order Calder\'{o}n
commutators, maximally modulated multilinear singular integrals, and
$q$-variation of $\omega$-Calder\'{o}n-Zygmund operators.
