  Multiband transport in superconductors is interesting both from an academic
as well as an application point of view. It has been postulated that interband
scattering can significantly affect the carrier dynamics in these materials. In
this article we present a detailed study of the electrical transport properties
of the high-mobility two-dimensional electron gas residing at the interface of
LaAlO$_3$/SrTiO$_3$, a prototypical multi-band superconductor. We show, through
careful measurements of the gate dependence of the magnetoresistance and
resistance fluctuations at ultra-low temperatures, that transport in the
superconducting regime of this system has contributions from two bands which
host carriers of very different characters. We identify a gate-voltage tunable
Lifshitz transition in the system and show that the resistance fluctuations
have strikingly different features on either side of it. At low carrier
densities, resistance noise is dominated by number-density fluctuations arising
from trapping-detrapping of charge carriers from defects in the underlying
SrTiO$_3$ substrate, characteristic of a single-band semiconductor. Above the
Lifshitz transition, the noise presumably originates from inter-band
scattering. Our work highlights the importance of inter-band scattering
processes in determining the transport properties of low-dimensional systems
and projects resistance fluctuation spectroscopy as a viable technique for
probing the charge carrier dynamics across a Lifshitz transition.
