  We consider the $\alpha$-$T_3$ model which provides a smooth crossover
between the honeycomb lattice with pseudospin $1/2$ and the dice lattice with
pseudospin $1$ through the variation of a parameter $\alpha$. We study the
dynamics of a wave packet representing a quasiparticle in the $\alpha$-T$_3$
model with zero and finite transverse magnetic field. For zero field, it is
shown that the wave packet undergoes a transient $zitterbewegung$ (ZB). Various
features of ZB depending on the initial pseudospin polarization of the wave
packet have been revealed. For an intermediate value of the parameter $\alpha$
i.e. for $0<\alpha<1$ the resulting ZB consists of two distinct frequencies
when the wave packet was located initially in $rim$ site. However, the wave
packet exhibits single frequency ZB for $\alpha=0$ and $\alpha=1$. It is also
unveiled that the frequency of ZB corresponding to $\alpha=1$ gets exactly half
of that corresponding to the $\alpha=0$ case. On the other hand, when the
initial wave packet was in $hub$ site, the ZB consists of only one frequency
for all values of $\alpha$. Using stationary phase approximation we find
analytical expression of velocity average which can be used to extract the
associated timescale over which the transient nature of ZB persists. On the
contrary the wave packet undergoes permanent ZB in presence of a transverse
magnetic field. Due to the presence of large number of Landau energy levels the
oscillations in ZB appear to be much more complicated. The oscillation pattern
depends significantly on the initial pseudospin polarization of the wave
packet. Furthermore, it is revealed that the number of the frequency components
involved in ZB depends on the parameter $\alpha$.
