We show rational homological stability for the classifying spaces of the monoid of homotopy self-equivalences and the block diffeomorphism group of iterated connected sums of products of spheres. The spheres can have different dimensions, but need to satisfy a certain connectivity assumption. The main theorems of this paper extend homological stability results for automorphism spaces of connected sums of products of spheres of the same dimension by Berglund and Madsen.
"On rational homological stability for block automorphisms of connected sums of products of spheres." Algebr. Geom. Topol. 19 (7) 3359 - 3407, 2019. https://doi.org/10.2140/agt.2019.19.3359