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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 (∼60ks each) about once every 4 days, and includes a reanalysis of previous XMM-Newton and Chandra observations. We aim at understanding the origin and location of the Fe K emission and absorption regions. We combine the results of time-resolved spectral analysis on both short and long time-scales including model-independent rms spectra. Mrk 509 shows a clear (EW=58±4eV) neutral Fe Kα emission line that can be decomposed into a narrow (σ=0.027keV) component (found in the Chandra HETG data) plus a resolved (σ=0.22keV) component. We find the first successful measurement of a linear correlation between the intensity of the resolved line component and the 3-10keV 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≲10³r_g) from the black hole (BH). The lack of a redshifted wing in the line poses a lower limit of ≥40r_g 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∼8-20eV) 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 r_g from the BH. In the latter interpretation, the presence of an ionised standard α-disc, down to a few r_g, 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. 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∼40-1000r_g from the BH.