In this paper, we characterize the F-transitive and the dF-transitive families of composition operators on Lp(X,B,μ), where (X,B,μ) is a σ-finite measure space. In particular, both necessary and sufficient conditions on the inducing mappings for F-transitive and dF-transitive composition operators are presented, which includes both Theorem 1.1 and Theorem 1.2 given in [1] in a more general frame. Moreover, we provide the characterizations of F-transitive and dF-transitive semigroups induced by semiflows on Lρp(X,K). And as a result, the sufficient condition of d-transitive semigroups induced by semiflows on Lρp(Ω,K) appears to be necessary, which answers a recent question posed by Kostić in [16]. Further, we provide the characterizations on the weight function ρ for F-transitive translation semigroups on Lρp(Δ,K), which extends the recent results in [10] and [13]. Some related examples of dF-transitive composition operators and C0-semigroups that are the solution semigroups of certain Cauchy problems are provided. In particular, we present the properties of dF-transitive composition operators on Lp(T,B,λ) induced by automorphisms on the complex unit disk D and show the equivalence of d-transitivity and d-mixing property for these operators.