Recent X-ray observations show absorbing winds with velocities up to mildly relativistic values of the order of ∼0.1c in a limited sample of six broad-line radio galaxies. They are observed as blueshifted Fe xxv–xxvi K-shell absorption lines, similarly to the ultrafast outflows (UFOs) reported in Seyferts and quasars. In this work we extend the search for such Fe K absorption lines to a larger sample of 26 radio-loud active galactic nuclei (AGN) observed with XMM–Newton and Suzaku. The sample is drawn from the Swift Burst Alert Telescope 58-month catalogue and blazars are excluded. X-ray bright Fanaroff–Riley Class II radio galaxies constitute the majority of the sources. Combining the results of this analysis with those in the literature we find that UFOs are detected in >27 per cent of the sources. However, correcting for the number of spectra with insufficient signal-to-noise ratio, we can estimate that the incidence of UFOs is this sample of radio-loud AGN is likely in the range f ≃ (50 ± 20) per cent. A photoionization modelling of the absorption lines with xstar allows us to estimate the distribution of their main parameters. The observed outflow velocities are broadly distributed between v_out ≲ 1000 km s⁻¹ and v_out ≃ 0.4c, with mean and median values of v_out ≃ 0.133c and v_out ≃ 0.117c, respectively. The material is highly ionized, with an average ionization parameter of logξ ≃ 4.5 erg s⁻¹ cm, and the column densities are larger than N_H > 10²² cm⁻². Overall, these characteristics are consistent with the presence of complex accretion disc winds in a significant fraction of radio-loud AGN and demonstrate that the presence of relativistic jets does not preclude the existence of winds, in accordance with several theoretical models.