In this paper, we calculate the vector, axial-vector and tensor form factors of Prightarrow T transition within the standard light-front (SLF) and covariant light-front (CLF) quark models (QMs). The self-consistency and Lorentz covariance of CLF QM with two types of correspondence schemes are investigated. The zero-mode effects and the spurious omega -dependent contributions to the form factors of Prightarrow T transition are analyzed. Employing a self-consistent CLF QM, we present our numerical predictions for the vector, axial-vector and tensor form factors of crightarrow (q,s) (q=u,d) induced D rightarrow (a_2,K^*_2), D_s rightarrow (K^*_2,f'_{2}), eta _c(1S) rightarrow (D^*_2,D^*_{s2}), B_c rightarrow (B^*_2,B^*_{s2}) transitions and brightarrow (q,s,c) induced B rightarrow (a_2,K^*_2,D^*_2), B_s rightarrow (K^*_2,f'_2,D^*_{s2}), B_c rightarrow (D^*_2,D^*_{s2},chi _{c2}(1P)), eta _b(1S) rightarrow (B^*_2,B^*_{s2}) transitions. Finally, in order to test the obtained form factors, the semileptonic Brightarrow {bar{D}}_2^*(2460)ell ^+nu _ell (ell =e,mu ) and {bar{D}}_2^*(2460)tau ^+nu _{tau } decays are studied. It is expected that our results for the form factors of Prightarrow T transition can be applied further to the relevant phenomenological studies of meson decays.