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The 𝛾-ray strength functions (GSFs) and nuclear level densities (NLDs) below the neutron threshold have been extracted for 111–113,116–122,124Sn from particle-𝛾 coincidence data with the Oslo method. The evolution of bulk properties of the low-lying electric dipole response has been investigated on the basis of the Oslo GSF data and results of a recent systematic study of electric- and magnetic dipole strengths in even-even Sn isotopes with relativistic Coulomb excitation. The obtained GSFs reveal a resonance-like peak on top of the tail of the isovector giant dipole resonance centered at ≈8 MeV and exhausting ≈2% of the classical Thomas-Reiche-Kuhn (TRK) sum. For mass numbers ≥118 the data suggest also a second peak centered at ≈6.5 MeV. It corresponds to 0.1%–0.5% of the TRK sum rule and shows an approximate linear increase with the mass number. In contrast with predictions of the relativistic quasiparticle random-phase and time-blocking approximation calculations, no monotonic increase in the total low-lying 𝐸⁢1 strength was observed in the experimental data from 111Sn to 124Sn, demonstrating rather similar strength distributions in these nuclei. The Oslo GSFs and NLDs were further used as inputs to constrain the cross sections and Maxwellian-averaged cross sections of (𝑛,𝛾) reactions in the Sn isotopic chain using talys. The obtained results agree well with other available experimental data and the recommended values from the JINA REACLIB, BRUSLIB, and KADoNiS libraries. Despite relatively small exhausted fractions of the TRK sum rule, the low-lying electric dipole strength makes a noticeable impact on the radiative neutron-capture cross sections in stable Sn isotopes. Moreover, the experimental Oslo inputs for the 121,123Sn ⁢(𝑛,𝛾)⁢ 122,124Sn reactions were found to affect the production of Sb in the astrophysical 𝑖 process, providing new constraints on the uncertainties of the resulting chemical abundances from multizone low-metallicity asymptotic giant branch stellar models.