The McKinsey axiom (M)□◊p→◊□p has a local first-order correspondent on the class of all weakly transitive frames WT. It globally corresponds to Lemmon’s condition (m∞) on WT. The formula (M) is canonical over the weakly transitive modal logic wK4=K⊕p∧□p→□□p. The modal logic wK4.1=wK4⊕M has the finite model property. The modal logics wK4.1T0n (n>0) form an infinite descending chain in the interval [wK4.1,K4.1] and each of them has the finite model property. Thus all the modal logics wK4.1 and wK4.1T0n (n>0) are decidable.