We present structure-preserving schemes on two-dimensional multipatch spline spaces with local tensor-product parametrization and non-matching interfaces. Our construction covers the case of shape-regular patches with different mappings and refined patches, under the assumption that neighbouring patches have nested resolutions. Following the broken-FEEC approach, our schemes involve de Rham sequences which are fully discontinuous across patch interfaces and conforming projection operators that enforce the minimal conformity constraints for the global de Rham spaces. In this talk, we describe the construction of these operators, state theoretical results regarding stability and structure-preservation and present applications to Poisson's and Maxwell's equations.