Generic coordinates transformation to a free falling frame

Taking a spherically symmetric metric of an arbitrary form:

$ds^2 = - A(r) dt^2 +B(r) dr^2 + C(r) [r^2d\theta^2 + r^2 \sin^2\theta d\phi^2] ~~~~~~~(1)$

where A,B,C are functions.

I would like to understand the steps to transforme (1) in a coordinate system $({\tau,\rho,\theta.\phi})$ where $\tau$ is the proper time of the free falling observer, and $\rho$ his constant in is frame. What Im looking for is a generalization of the Lemaître coordinates (Wikipedia page of the Lemaître coordinates) for arbitrary static solution of the Einstein field equations that is well defined outside the horizon.

NB: I'm trying to make this question as general as possible but to begin with, I do not totally understand how one can go from Schwarzschild to Lemaître. For simplicity one can set: $A=B^{-1},C=1$.