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We present a detailed study of the stellar kinematic properties of red member galaxies in the cores of four strong lensing galaxy clusters at intermediate redshift included in the the Hubble Frontier Fields programme: Abell 2744 ($z=0.307$), Abell S1063 ($z=0.346$), MACS J0416.1$-$2403 ($z=0.397$), and MACS J1149.6$+$2223 ($z=0.542$). We focussed on a sample of 723 red cluster members in the four clusters and we measured their structural parameters using MORPHOFIT for all Hubble Frontier Fields bands. Taking advantage of deep (3.1h to 17h of exposure) integral-field spectroscopy from MUSE on the Very Large Telescope, we tested a pipeline based on the public spectral fitting code pPXF to systematically measure the line-of-sight stellar velocity dispersion $\sigma$ of cluster members with a spectral $S/N\geq 10$, with a statistical uncertainty consistently below 5%. The resulting catalogue contains 213 measured $\sigma$ values across the four clusters. We calibrated the Fundamental Plane relation in the rest-frame $r$ band for the early-type cluster members, selected from their colour and morphology, finding compatible parameters both across the clusters, and noting hints of zero-point evolution with redshift. Finally, we used the calibrated Fundamental Plane relations to assign a velocity dispersion value to all 723 red cluster members and studied the velocity dispersion function for each cluster, down to $\log \sigma \, \mathrm{[km \, s^{-1}] = 1.5}$. A Schechter-function fit of the velocity functions suggests compatible parameters: a positive $\alpha$ slope with values in the range $0.55-1.60$, and $\log\sigma^*\, [\mathrm{km\,s^{-1}}]$ between $2.18$ and $2.47$. Compared to previous works, we extend the systematic study of the central velocity dispersion of cluster galaxies to lower-$\sigma$ regimes.
