Equation of Motion from Lagrangian: Expand the Lagrangian by orders of perturbation to get higher order equation of motion?

Just to state this first: I am not asking about higher order derivatives, so no jerk, snap, crackle or pop is involved here. I am asking about higher order perturbative equations of motion from an expanded Lagrangian.

Say, I have a Lagrangian, depending on a scalar field $phi$ and a metric $g_{ij}$, which I perturb to get
$$
L(phi,g_{ij}) = L_0 + L_1 + L_2 +…
phirightarrowphi(t)+deltaphi(t,x,y,z)
g_{ij}rightarrow g_{ij}+delta g_{ij}
$$
where $L_1$ is of order $epsilon$, $L_2$ is of order $epsilon^2$, and so forth, with $epsilon$ the order of the perturbation.

Now, I would like to derive equations of motion. The background equation of motion is given by the Euler-Lagrange equation
$$
frac{partial L_0}{partial phi}-frac{mathrm{d}}{mathrm{d}t}frac{partial L_0}{partial dot{phi}}=0.
$$
But how do I get, say, the first order equations of motion?