  We have characterized the pulsation properties of 164 candidate RR Lyrae
variables (RRLs) and 55 candidate Anomalous and/or short-period Cepheids in Leo
I dwarf spheroidal galaxy. On the basis of its RRLs Leo I is confirmed to be an
Oosterhoff-intermediate type galaxy, like several other dwarfs. We show that in
their pulsation properties, the RRLs 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
(~1300stars) with those in the Galactic halo field (~14,000stars) and globular
clusters (~1000stars). Field RRLs show a significant change in their period
distribution when moving from the inner (dG<14 kpc) to the outer (dG>14kpc)
halo regions. This suggests that the halo formed from (at least) two dissimilar
progenitors or types of progenitor. Considered together, the RRLs 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 RRLs
in globular clusters. In particular, the samples of fundamental-mode RRLs in
dwarfs seem to lack High Amplitudes and Short Periods ("HASP":AV>1.0mag and P
<0.48d) when compared with those observed in the Galactic halo field and
globular clusters. The observed properties of RRLs 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.
