Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin--Daykin--Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell--Felsner--Trotter. The proofs are based on a multivariate generalization of the Ahlswede--Daykin inequality.