Electronic and optical properties of materials are affected by atomic motion through the electron-phonon interaction. This interaction has two noticeable effects on the electronic structure: variation of band gaps as a function of temperature, but also band-gap renormalization even at absolute zero temperature, due to zero-point motion, directly linked to polaron binding energies. Ignored in most calculations of the electronic structure, zero-point effects have been evaluated recently from first principles for several materials. Many of these calculations relied on the adiabatic approximation, reasonably valid for materials without infrared activity, but eagerly applied to other materials. We present the first large-scale (29 materials) first-principles evaluation of the zero-point renormalization of band edges beyond the adiabatic approximation. For materials with light elements, the band gap renormalization is often larger than 0.3 eV, and up to 1.1 eV. This effect cannot be ignored if accurate band gaps are sought. For infrared-active materials, global agreement with available experimental data is obtained only when dynamical effects are taken into account. They even dominate zero-point renormalization for many materials. We present a generalized Fröhlich model that represents accurately the dynamical contributions, and assess its accuracy against first-principles results. This model describes the essential physics and it accounts for more than half of the total ZPR for a large set of materials, especially for the valence band edges, despite its neglect of interband electronic transitions, Debye-Waller contribution and acoustic phonon contributions present in the full first-principles approach. By the same token, the domain of validity of the hypotheses underlying the Fröhlich model, used for decades, is established.