We introduce a new family of models for spin-glass systems. This family has quenched site dilution with varying strengths of local correlations and modulated ordered phases in the pure limit. The correlated dilution mimics the experimental situation in Cu-Mn, Au-Fe, and Pt-Mn. Relatively weak correlations are conjectured to be sufficient to explain the observed interactions between regions of modulated spin order and coexisting smaller ferromagnetic regions in CuMn. The models are investigated by the transmission-of-order approach, which relates the zero-temperature line of the temperature-dilution phase diagram to a family of correlated percolation models. Exact results for the correlated percolation models on the Cayley tree and simulations on the square lattice are presented. We find novel phases in which percolation occurs on some but not all sublattices.