From archival ground-based images of the Leo I dwarf spheroidal galaxy, we have identified and characterized the pulsation properties of 164 candidate RR Lyrae variables and 55 candidate anomalous and/or short-period Cepheids. We have also identified 19 candidate long-period variable stars and 13 other candidate variables whose physical nature is unclear, but due to the limitations of our observational material we are unable to estimate reliable periods for them. On the basis of its RR Lyrae star population, Leo I is confirmed to be an Oosterhoff-intermediate type galaxy, like several other dwarf spheroidals. From the RR Lyrae stars we have derived a range of possible distance moduli for Leo I : 22.06 ± 0.08 ≲ μ0 ≲ 22.25 ± 0.07 mag depending on the metallicity assumed for the old population ([Fe/H] from -1.43 to -2.15). This is in agreement with previous independent estimates. We show that in their pulsation properties, the RR Lyrae stars-representing the oldest stellar population in the galaxy-are not significantly different from those of five other nearby, isolated dwarf spheroidal galaxies. A similar result is obtained when comparing them to RR Lyrae stars in recently discovered ultra-faint dwarf galaxies. We are able to compare the period distributions and period-amplitude relations for a statistically significant sample of ab-type RR Lyrae stars in dwarf galaxies (˜1300 stars) with those in the Galactic halo field (˜14,000 stars) and globular clusters (˜1000 stars). Field RRLs show a significant change in their period distribution when moving from the inner (dG ≲ 14 kpc) to the outer (dG ≳ 14 kpc) halo regions. This suggests that the halo formed from (at least) two dissimilar progenitors or types of progenitor. Considered together, the RR Lyrae stars in classical dwarf spheroidal and ultra-faint dwarf galaxies-as observed today-do not appear to follow the well defined pulsation properties shown by those in either the inner or the outer Galactic halo, nor do they have the same properties as RR Lyraes in globular clusters. In particular, the samples of fundamental-mode RR Lyrae stars in dwarf galaxies seem to lack High Amplitudes and Short Periods ("HASP": AV≥1.0 mag and P ≲ 0.48 d) when compared with those observed in the Galactic halo field and globular clusters. The observed properties of RR Lyrae stars do not support the idea that currently existing classical dwarf spheroidal and ultra-faint dwarf galaxies are surviving representative examples of the original building blocks of the Galactic halo.