  We consider all totally asymmetric simple exclusion processes (TASEPs) whose
transition probabilities are given in the Sch\"utz-type formulas and which jump
with homogeneous rates. We show that the multi-point distribution of particle
positions and the coefficient of KPZ scaling are described using the
probability generating function of the distribution followed when the rightmost
particle jumps. For all TASEPs satisfying certain assumptions, We also prove
the pointwise convergence of the kernels appearing in the joint distribution of
particle positions to those appearing in the KPZ fixed point formula. Our
result generalizes the result of Matetski, Quastel, and Remenik [Acta Math.,
227, 115-203, (2021)].
