Numerous physical phenomena, including the flow in fractured porous media, the analysis of fiber-reinforced materials, or the simulation of small circulatory systems in living tissues, involve models defined on separate spatial scales and can be characterized as interlinked partial differential equations [1,2]. These equations are defined within domains of varying dimensions that interpenetrate each other and are generally called mixed-dimensional partial differential equations. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge [3,4]. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers [4]. We address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem, and we discuss the inf-sup stability of the proposed numerical scheme. From the perspective of mixed-dimensional problem theory, we provide general mathematical tools to construct interconnected problems spanning different dimensions. Regarding numerical approximation, we demonstrate the relationship among the mesh size, the dimension of the Lagrange multiplier space, and the inclusion size in typical setups relevant for practical use. This analysis is further enhanced by illustrative numerical examples.