  Models with dust reference fields in relational formalism have proved useful
in understanding the construction of gauge-invariant perturbation theory to
arbitrary orders in the canonical framework. These reference fields modify the
dynamical equations for perturbation equations. However, important questions
remain open on the relation with conventional perturbation theories of inflaton
coupled to gravity and of multi-fluid systems, and on understanding
modifications in terms of physical degrees of freedom. These gaps are filled in
this manuscript for Brown-Kucha\v{r} and Gaussian dust models, both of which
involve three scalar physical degrees of freedom. We establish a relationship
of these models with conventional inflationary and multi-fluid system of
inflation and ordinary dust by introducing a set of gauge invariant variables
on the reduced phase space of the dust reference models. We find the
modifications due to dust clocks to Bardeen equation in the longitudinal gauge
and Mukhanov-Sasaki equation in the spatially-flat gauge, in terms of physical
degrees of freedom. This results in a closed system of equations for all the
degrees of freedom needed to explore the evolution of the scalar perturbations.
Our analysis shows for the first time that even for two-fluid systems, there is
a natural choice of the set of gauge invariant variables for each chosen gauge
which not only offers a direct physical interpretation but also results in
simplifications to the dynamical equations.
