  In this paper, we investigate the tensor form factors of $P\to P,\,S,\,V$ and
$A$ transitions within the standard light-front (SLF) and the covariant
light-front (CLF) quark models (QMs). The self-consistency and Lorentz
covariance of CLF QM are analyzed via these quantities, and the effects of
zero-mode are discussed. For the $P\to V$ and $A$ transitions, besides the
inconsistence between the results extracted via longitudinal and transverse
polarization states, which is caused by the residual $\omega$-dependent
spurious contributions, we find and analyze a "new" self-consistence problem of
the traditional CLF QM, which is caused by the different strategies for dealing
deal with the trace term in CLF matrix element. A possible solution to the
problems of traditional CLF QM is discussed and confirmed numerically. Finally,
the theoretical predictions for the tensor form factors of some $c\to q,\,s$
and $b\to q,\,s\,,c$ ($q=u,d$) induced $P\to P,\,S,\,V$ and $A$ transitions are
updated within the CLF QM with a self-consistent scheme.
