So first of all, there are many people who have investigated various aspects of asymptotically flat space-times and this answer is not an attempt at a review of all of those works, it is more an assorted reading list. To your list of names that contributed to the field you should definitely add people such as Pirani, Metzner, Thorne, Geroch, Hansen, the Arnowitt-Deser-Misner trio, and in more specific contexts also Blanchet, Damour, or Strominger.
Stationary asymptotically flat space-times are quite straightforward, they have sets of multipoles and that is basically it. However, what is more interesting are general, dynamical space-times, in which gravitational radiation must generically be emitted. So, as you will see, the theory of asymptotically flat space-times is intimately tied to theory of gravitational radiation.
An assorted historical list
A first highly influential formalism used to asymptotically flat space-times at spatial infinity is the Arnowitt-Deser-Misner (ADM) formalism scattered around 4 papers published in 1960. Fortunately, they also published a review:
The funny thing is that the primary aim of ADM was not to study the asymptotically flat structure, but rather to understand the dynamical variables or relativity, but they end up discussing the issue.
Skipping some other developments, another must-read for the structure of asymptotically flat space-time is
In order to understand it thoroughly, it is also good to read the previous paper in the series by
where the constructions are done in axi-symmetric space-times but are more complete and example oriented (and also this paper is more careful to lay out its assumptions). The genius of these two papers is to clearly show that gravitational waves can carry away the mass of the radiating system thanks to the non-linearities of the equations even though the character of the radiation is never monopolar. This is an example of a non-linear memory effect of gravitational radiation. The papers also include the first statement of the group nowadays known as the Bondi-Metzner-Sachs (BMS) group.
Both the BMS and the ADM formalism are ultimately interested in the boundary conditions and the initial value problem, and it might not be entirely obvious what is the difference between them. The difference has been elucidated mainly by Penrose in:
In this article he makes obvious the difference between the various asymptotic regions such as the null and spatial infinities and essentially summarizes the previous results in this new language.
Next to mention is
In this paper, Ashtekar and Hansen provide covariant definitions of spatial infinity and null infinity and document the asymptotic symmetries of spatial infinity for the first time.
One last must-read on memory and asymptotically flat infinity is a recent development by
It documents how the passage of a gravitational wave through infinity induces a memory effect that can be understood as a BMS transformation of far-away observers. Additionally, it relates this effect to Weinberg's soft theorems for gravitons.
Asymptotic multipole structure
Last but not least, I would also like to mention two more "pragmatic" works on asymptotics of gravitational fields in the context of gravitational radiation:
The work of Thorne is most important because it shows the quadrupolar nature of gravitational radiation for arbitrary sources by asymptotic analysis. (Also its Box 1 gives a nice review of literature.) However, it also provides a multipolar expansion that should in principle converge not only asymptotically far but also relatively close to the source. The works of Blanchet & Damour then pushes the (Bonnor-)Thorne approach into a systematic formalism that is nowadays used in the computation of gravitational-wave inspirals.
Some recent reviews
On the null structure and BMS, I recommend the reviews
For a taste of how the asymptotic form of the metric is used to compute the emission of gravitational waves, see