  We have performed systematic investigations of transport through graphene on
hexagonal boron nitride (hBN) substrates, together with confocal Raman
measurements and a targeted theoretical analysis, to identify the dominant
source of disorder in this system. Low-temperature transport measurements on
many devices reveal a clear correlation between the carrier mobility $\mu$ and
the width $n^*$ of the resistance peak around charge neutrality, demonstrating
that charge scattering and density inhomogeneities originate from the same
microscopic mechanism. The study of weak-localization unambiguously shows that
this mechanism is associated to a long-ranged disorder potential, and provides
clear indications that random pseudo-magnetic fields due to strain are the
dominant scattering source. Spatially resolved Raman spectroscopy measurements
confirm the role of local strain fluctuations, since the line-width of the
Raman 2D-peak --containing information of local strain fluctuations present in
graphene-- correlates with the value of maximum observed mobility. The
importance of strain is corroborated by a theoretical analysis of the relation
between $\mu$ and $n^*$ that shows how local strain fluctuations reproduce the
experimental data at a quantitative level, with $n^*$ being determined by the
scalar deformation potential and $\mu$ by the random pseudo-magnetic field
(consistently with the conclusion drawn from the analysis of
weak-localization). Throughout our study, we compare the behavior of devices on
hBN substrates to that of devices on SiO$_2$ and SrTiO$_3$, and find that all
conclusions drawn for the case of hBN are compatible with the observations made
on these other materials. These observations suggest that random strain
fluctuations are the dominant source of disorder for high-quality graphene on
many different substrates, and not only on hexagonal boron nitride.
