  We present new measurements of the free-streaming of warm dark matter (WDM)
from Lyman-$\alpha$ flux-power spectra. We use data from the medium resolution,
intermediate redshift XQ-100 sample observed with the X-shooter spectrograph
($z=3 - 4.2$) and the high-resolution, high-redshift sample used in Viel et al.
(2013) obtained with the HIRES/MIKE spectrographs ($z=4.2 - 5.4$). Based on
further improved modelling of the dependence of the Lyman-$\alpha$ flux-power
spectrum on the free-streaming of dark matter, cosmological parameters, as well
as the thermal history of the intergalactic medium (IGM) with hydrodynamical
simulations, we obtain the following limits, expressed as the equivalent mass
of thermal relic WDM particles. The XQ-100 flux power spectrum alone gives a
lower limit of 1.4 keV, the re-analysis of the HIRES/MIKE sample gives 4.1 keV
while the combined analysis gives our best and significantly strengthened lower
limit of 5.3 keV (all 2$\sigma$ C.L.). The further improvement in the joint
analysis is partly due to the fact that the two data sets have different
degeneracies between astrophysical and cosmological parameters that are broken
when the data sets are combined, and more importantly on chosen priors on the
thermal evolution. These results all assume that the temperature evolution of
the IGM can be modelled as a power law in redshift. Allowing for a non-smooth
evolution of the temperature of the IGM with sudden temperature changes of up
to 5000K reduces the lower limit for the combined analysis to 3.5 keV. A WDM
with smaller thermal relic masses would require, however, a sudden temperature
jump of $5000\,K$ or more in the narrow redshift interval $z=4.6-4.8$, in
disagreement with observations of the thermal history based on high-resolution
resolution Lyman-$\alpha$ forest data and expectations for photo-heating and
cooling in the low density IGM at these redshifts.
