  Much having learned about Floquet dynamics of pseudospin-$1/2$ system namely,
graphene, we here address the stroboscopic properties of a periodically kicked
{three-band fermionic system such as $\alpha$-T$_3$ lattice. This particular
model provides an interpolation between graphene and dice lattice via the
continuous tuning of the parameter $\alpha$ from 0 to 1.} In the case of dice
lattice ($\alpha=1$), we reveal that one can, in principle, engineer various
types of low energy dispersions around some specific points in the Brillouin
zone by tuning the kicking parameter in the Hamiltonian along a particular
direction. Our analytical analysis shows that one can experience different
quasienergy dispersions for example, Dirac type, semi-Dirac type, gapless line,
absolute flat quasienergy bands, depending on the specific values of the
kicking parameter. Moreover, we numerically study the dynamics of a wave packet
in dice lattice. The quasienergy dispersion allows us to understand the
instantaneous structure of wave packet at stroboscopic times. We find a
situation where absolute flat quasienergy bands lead to a complete dynamical
localization of the wave packet. {Aditionally, we calculate the quasienergy
spectrum numerically for $\alpha$-T$_3$ lattice. A periodic kick in a
perpendicular (planar) direction breaks (preserves) the particle-hole symmetry
for $0<\alpha<1$. Furthermore, it is also revealed that the dynamical
localization of wave packet does not occur at any intermediate $\alpha \ne
0,\,1$.}
