  Mean-field games with absorption is a class of games, that have been
introduced in Campi and Fischer (2018) and that can be viewed as natural limits
of symmetric stochastic differential games with a large number of players who,
interacting through a mean-field, leave the game as soon as their private
states hit some given boundary. In this paper, we push the study of such games
further, extending their scope along two main directions. First, we allow the
state dynamics and the costs to have a very general, possibly
infinite-dimensional, dependence on the (non-normalized) empirical
sub-probability measure of the survivors' states. This includes the
particularly relevant case where the mean-field interaction among the players
is done through the empirical measure of the survivors together with the
fraction of absorbed players over time. Second, the boundedness of coefficients
and costs has been considerably relaxed including drift and costs with linear
growth in the state variables, hence allowing for more realistic dynamics for
players' private states. We prove the existence of solutions of the MFG in
strict as well as relaxed feedback form, and we establish uniqueness of the MFG
solutions under monotonicity conditions of Lasry-Lions type. Finally, we show
in a setting with finite-dimensional interaction that such solutions induce
approximate Nash equilibria for the $N$-player game with vanishing error as
$N\rightarrow \infty$.
