A foundational challenge in uncertainty quantification involves estimating a probability measure on the space of uncertain parameters such that its push-forward through a computational model matches an observed probability measure on the output data associated with quantities of interest (QoI). When multiple, distinct sets of observational data are available, the desired parameter measure should simultaneously satisfy multiple push-forward constraints associated with various subsets of the QoI. In this work, we present a convergent measure-theoretic framework for solving this problem based on an iterative application of Data-Consistent Inversion (DCI). We first rigorously establish the theoretical optimality of the DCI solution to the standard problem, proving that it minimizes the $f$-divergence over the space of all possible pullback measures that satisfy the push-forward constraint. This optimality property provides the foundation for our iterative DCI scheme, which is shown to converge to a solution of the multiple push-forward constraint problem. This iterative solution minimizes the cumulative $f$-divergence across all constraints and, under uniform initializations, represents the maximal entropy solution (the I-projection) onto the intersection of the solution sets. We provide a rigorous convergence analysis for the proposed method and demonstrate its practical utility through numerical examples, including a high-dimensional parameter space governed by partial differential equations, where the iterative approach robustly avoids the complexities associated with approximating high-dimensional joint observed measures.