We use deep Herschel observations of the complete 2Jy sample of powerful radio active galactic nuclei (AGNs) in the local Universe (0.05 < z < 0.7) to probe their cool interstellar medium (ISM) contents and star-forming properties, comparing them against other samples of nearby luminous AGNs and quiescent galaxies. This allows us to investigate triggering and feedback mechanisms. We find that the dust masses of the strong-line radio galaxies (SLRGs) in our sample are similar to those of radio-quiet quasars, and that their median dust mass (M_dust_ = 2 * 107 M_{sun}_) is enhanced by a factor of ~200 compared to that of non-AGN ellipticals, but lower by a factor of ~16 relative to that of local ultraluminous infrared galaxies (ULIRGs). Along with compelling evidence for merger signatures in optical images, the SLRGs in our sample also show relatively high star formation efficiencies, despite the fact that many of them fall below the main sequence for star-forming galaxies. Together, these results suggest that most of our SLRGs have been retriggered by late-time mergers that are relatively minor in terms of their gas contents. In comparison with the SLRGs, the radio AGNs with weak optical emission lines (weak-line radio galaxies - WLRGs) and edge-darkened radio jets (Fanaroff-Riley Class I radio sources - FRIs) have both lower cool ISM masses and star formation rates (by a factor of >30), consistent with being fuelled by a different mechanism (e.g. the direct accretion of hot gas).