We will recall a question published in E. Glasner's paper \emph{On a question of Kazhdan and Yom Din} regarding the possibility to approximate functionals on a Banach space which are almost invariant with respect to an action of a discrete group by functionals that are invariant. Then the case when the Banach space is a Lipschitz-free space equipped with an action induced by an action by isometries on the underlying space will be discussed. We will mention a few different conditions sufficient for the answer to be positive; for example the case of free or finitely presented groups endowed with left-invariant metrics acting on themselves by translations. Relations to quasimorphisms will also be briefly mentioned.