The discrete collisional breakage equation, which captures the dynamics of cluster growth when clusters undergo binary collisions with possible matter transfer, is discussed in this article. The existence of global massconserving solutions is investigated for the collision kernels ai,j = A(i αj β +i β j α), i, j ≥ 1, with α ∈ (−∞, 1), β ∈ [α, 1] ∩ (0, 1], and A > 0 and for a large class of possibly unbounded daughter distribution functions. All algebraic superlinear moments of these solutions are bounded on time intervals [T,∞) for any T > 0. The uniqueness issue is further handled under additional restrictions on the initial data. Finally, non-trivial stationary solutions are constructed by a dynamical approach.