The most intensively star-forming galaxies are extremely luminous at far-infrared (FIR) wavelengths, highly obscured at optical and ultraviolet wavelengths, and lie at z ≥ 1-3. We present a programme of Herschel FIR spectroscopic observations with the SPIRE FTS and photometric observations with PACS, both on board Herschel, towards a sample of 45 gravitationally lensed, dusty starbursts across z ∼ 1-3.6. In total, we detected 27 individual lines down to 3-σ, including nine [C II] 158-µm lines with confirmed spectroscopic redshifts, five possible [C II] lines consistent with their far-infrared photometric redshifts, and in some individual sources a few [O III] 88-µm, [O III] 52-µm, [O I] 145-µm, [O I] 63-µm, [N II] 122-µm, and OH 119-µm (in absorption) lines. To derive the typical physical properties of the gas in the sample, we stack all spectra weighted by their intrinsic luminosity and by their 500-µm flux densities, with the spectra scaled to a common redshift. In the stacked spectra, we detect emission lines of [C II] 158-µm, [N II] 122-µm, [O III] 88-µm, [O III] 52-µm, [O I] 63-µm, and the absorption doublet of OH at 119-µm, at high fidelity. We find that the average electron densities traced by the [N II] and [O III] lines are higher than the average values in local star-forming galaxies and ULIRGs, using the same tracers. From the [N II]/[C II] and [O I]/[C II] ratios, we find that the [C II] emission is likely dominated by the photo-dominated regions (PDR), instead of by ionised gas or large-scale shocks.