The author considers a large class of anisotropic flows for closed convex hypersurfaces in \({\mathbb{R}}^{n+1}\). These are the so-called mixed discriminant flows which he introduced in [B. Andrews, Commun. Anal. Geom. 2, 53-64 (1994; Zbl 0839.53049)], and include flows by positive powers of the Gauss curvature \(K\) or by positive powers of the ratio \(K/E_k\) where \(E_k\), \(1\leq k < n\), denotes the \(k\)-th elementary symmetric function of the principal curvatures. The main result is that for certain flows of this type, if a solution converges to a homothetic solution even in a very weak sense, or for a subsequence of times approaching the final time, then it must converge smoothly. In particular, this implies that there can be at most one homothetic solution of the flow. The main step of the proof is to show that certain integral quantities associated with the flow are monotone in time, for a certain range of powers of \(K\) or \(K/E_k\), respectively. For flows by sufficiently small positive powers of \(K\) it is known that there are multiple homothetic solutions.