  We develop an action principle to construct the dynamics that give rise to a
minimal generalization of Einstein's equations, where the speed of light ($c$),
the gravitational constant ($G$) and the cosmological constant ($\Lambda$) are
allowed to vary. Our construction preserves general covariance of the theory,
which yields a general dynamical constraint on $c$, $G$ and $\Lambda$. This
action is general and can be applied to describe different cosmological
solutions. We apply this formulation to the initial condition puzzles of the
early universe and show that it generates a dynamical mechanism to obtain the
homogeneous and flat universe we observe today. We rewrite the conditions
necessary to solve the horizon and flatness problems in this framework, which
does not necessarily lead to an accelerated expansion as in inflation. Then, we
show how the dynamics of the scalar field that represents $c$ or $G$ (and
$\Lambda$) can be used to solve the problems of the early universe cosmology by
means of different ways to c-inflate the horizon in the early universe. By
taking $\Lambda = 0$, we show that the dynamics of the scalar field
representing $c$ can be described once a potential is given.
