Summary: We prove various decay bounds on solutions \((f_{n} : n >0)\) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on \(n^{\ell}f_{n}\) in terms of a suitable average of the moments of the initial data for every positive \(\ell\). As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of \({L^p(\mathbb{R}^d \times [0, T])}\) norms of the moments \({X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}\), \(({\int_0^{\infty} m^a f_m(x, t)dm}\) in the case of continuous Smoluchowski’s equation) for every \({p \in [1, \infty]}\). In previous papers [the author, Proc. R. Soc. Edinb., Sect. A, Math. 140, No. 5, 1041–1059 (2010; Zbl 1206.35055); A. Hammond and the author, Commun. Math. Phys. 276, No. 3, 645–670 (2007; Zbl 1132.35090)] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient \(d(n)\) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function \({\phi(n)}\) that is closely related to the total increase of the diffusion coefficient in the interval \((0, n]\).