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This paper presents an analytical framework for the analysis of time-varying metal-based metamaterials. Concretely, we particularize the study to time-modulated metal-air interfaces embedded between two different semi-infinite media that are illuminated by monochromatic plane waves of frequency $ω_0$. The formulation is based on a Floquet-Bloch modal expansion, which takes into account the time periodicity of the structure ($T_s = 2π/ ω_s)$, and integral-equation techniques. It allows to extract the reflection/transmission coefficients as well as to derive nontrivial features about the dynamic response and dispersion curves of time-modulated metal-based screens. In addition, the proposed formulation has an associated analytical equivalent circuit that gives physical insight to the diffraction phenomenon. Similarities and differences between space- and time-modulated metamaterials are discussed via the proposed circuit model. Finally, some analytical results are presented to validate the present framework. A good agreement is observed with numerical computations provided by a self-implemented finite-difference time-domain (FDTD) method. Interestingly, the present results suggest that time-modulated metal-based screens can be used as pulsed sources (when $ω_s \ll ω_0$), beamformers ($ω_s \sim ω_0$) to redirect energy in specific regions of space, and analog samplers ($ω_s \gg ω_0$).