Context. We report on a detailed study of the Fe K emission/absorption complex in the nearby, bright Seyfert 1 galaxy Mrk 509. The study is part of an extensive XMM-Newton monitoring consisting of 10 pointings (~60 ks each) about once every 4 days, and includes a reanalysis of previous XMM-Newton and Chandra observations. Aims. We aim at understanding the origin and location of the Fe K emission and absorption regions. Results. Mrk 509 shows a clear (EW = 58 ± 4 eV) neutral Fe Kα emission line that can be decomposed into a narrow (σ = 0.027 keV) component (found in the Chandra HETG data) plus a resolved (σ = 0.22 keV) component. We find the first successful measurement of a linear correlation between the intensity of the resolved line component and the 3–10 keV flux variations on time scales of years down to a few days. The Fe Kα reverberates the hard X-ray continuum without any measurable lag, suggesting that the region producing the resolved Fe Kα component is located within a few light days to a week (r ≲ 103rg) from the black hole (BH). The lack of a redshifted wing in the line poses a lower limit of  ≥40 rg for its distance from the BH. The Fe Kα could thus be emitted from the inner regions of the BLR, i.e. within the  ~80 light days indicated by the Hβ line measurements. In addition to these two neutral Fe Kα components, we confirm the detection of weak (EW ~ 8–20 eV) ionised Fe K emission. This ionised line can be modelled with either a blend of two narrow Fe xxv and Fe xxvi emission lines (possibly produced by scattering from distant material) or with a single relativistic line produced, in an ionised disc, down to a few rg from the BH. In the latter interpretation, the presence of an ionised standard α-disc, down to a few rg, is consistent with the source high Eddington ratio. Finally, we observe a weakening/disappearing of the medium- and high-velocity high-ionisation Fe K wind features found in previous XMM-Newton observations. Conclusions. This campaign has made the first reverberation measurement of the resolved component of the Fe Kα line possible, from which we can infer a location for the bulk of its emission at a distance of r ~ 40–1000 rg from the BH.