We formulate a new theory for how caging constraints in glass-forming liquids at a surface or interface are modified and then spatially transferred, in a layer-by-layer bootstrapped manner, into the film interior in the context of the dynamic free energy concept of the Nonlinear Langevin Equation theory approach. The dynamic free energy at any mean location involves contributions from two adjacent layers where confining forces are not the same. At the most fundamental level of the theory, the caging component of the dynamic free energy varies essentially exponentially with distance from the interface, saturating deep enough into the film with a correlation length of modest size and weak sensitivity to thermodynamic state. This imparts a roughly exponential spatial variation of all the key features of the dynamic free energy required to compute gradients of dynamical quantities including the localization length, jump distance, cage barrier, collective elastic barrier and alpha relaxation time. The spatial gradients are entire of dynamical, not structural nor thermodynamic, origin. The theory is implemented for the hard sphere fluid and diverse interfaces which can be a vapor, a rough pinned particle solid, a vibrating pinned particle solid, or a smooth hard wall. Their basic description at the level of the spatially-heterogeneous dynamic free energy is identical, with the crucial difference arising from the first layer where dynamical constraints can be weakened, softened, or hardly changed depending on the specific interface. Numerical calculations establish the spatial dependence and fluid volume fraction sensitivity of the key dynamical property gradients for five different model interfaces. Comparison of the theoretical predictions for the dynamic localization length and glassy modulus with simulations and experiments for systems with a vapor interface reveals good agreement.
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