  We revisit and improve the analytic study arXiv:1804.03462 of spherically
symmetric but dynamical black holes in Einstein's gravity coupled to a real
scalar field. We introduce a series expansion in a small parameter $\epsilon$
that implements slow time dependence. At the leading order, the generic
solution is a quasi-stationary Schwarzschild-de Sitter (SdS) metric, i.e. one
where time-dependence enters only through the mass and cosmological constant
parameters of SdS. The two coupled ODEs describing the leading order time
dependence are solved up to quadrature for an arbitrary scalar potential.
Higher order corrections can be consistently computed, as we show by explicitly
solving the Einstein equations at the next to leading order as well. We comment
on how the quasi-stationary expansion we introduce here is equivalent to the
non-relativistic $1/c$ expansion.
