We study analytically the time-dependent boundary conditions of superconducting microwave circuit experiments in the high plasma frequency limit, in which the conditions are Robin-type and relate the value of the field to the spatial derivative of the field. We give an explicit solution to the field evolution for boundary condition modulations that are small in magnitude but may have arbitrary time dependence, in a formalism that applies both to a semiopen waveguide and to a closed waveguide with two independently adjustable boundaries. The correspondence between the microwave Robin boundary conditions and the mechanically moving Dirichlet boundary conditions of the dynamical Casimir effect is shown to break down at high field frequencies, approximately 1 order of magnitude above the frequencies probed in the 2011 experiment of Wilson et al. Our results bound the parameter regime in which a microwave circuit can be used to model relativistic effects in a mechanically moving cavity, and they show that beyond this parameter regime moving mirrors produce more particles and generate more entanglement than their nonmoving microwave waveguide simulations.