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Approximating functions of a large number of variables poses particular challenges often subsumed under the term ``Curse of Dimensionality'' (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be avoided only by exploiting some typically hidden {\em structural sparsity}. In this paper we propose a general framework for new model classes of functions in high dimensions. They are based on suitable notions of {\em compositional dimension-sparsity} quantifying, on a continuous level, approximability by compositions with certain structural properties. In particular, this describes scenarios where deep neural networks can avoid the CoD. The relevance of these concepts is demonstrated for {\em solution manifolds} of parametric transport equations. For such PDEs parameter-to-solution maps do not enjoy the type of high order regularity that helps to avoid the CoD by more conventional methods in other model scenarios. Compositional sparsity is shown to serve as the key mechanism forn proving that sparsity of problem data is inherited in a quantifiable way by the solution manifold. In particular, one obtains convergence rates for deep neural network realizations showing that the CoD is indeed avoided.