We study bridging transitions between a pair of nonplanar surfaces. We show that the transition can be described using a generalized Kelvin equation by mapping the system to a slit of finite length. The proposed equation is applied to analyze the asymptotic behavior of the growth of the bridging film, which occurs when the confining walls are gradually flattened. This phenomenon is characterized by a power-law divergence with geometry-dependent critical exponents that we determine for a wide class of walls' geometries. In particular, for a linear-wedge model, a covariance law revealing a relation between a geometric and Young's contact angle is presented. These predictions are shown to be fully in line with the numerical results obtained from a microscopic (classical) density functional theory.