  We study the zero temperature quenching dynamics of various extensions of the
transverse Ising model (TIM) when the transverse field is linearly quenched
from $-\infty$ to $+\infty$ (or zero) at a finite and uniform rate. The rate of
quenching is dictated by a characteristic scale given by $\tau$. The density of
kinks produced in these extended models while crossing the quantum critical
points during the quenching process is calculated using a many body
generalization of the Landau-Zener transition theory. The density of kinks in
the final state is found to decay as $\tau^{-1/2}$. In the first model
considered here, the transverse Ising Hamiltonian includes an additional
ferromagnetic three spin interaction term of strength $J_3$. For $J_3<0.5$, the
kink density is found to increase monotonically with $J_3$ whereas it decreases
with $J_3$ for $J_3>0.5$. The point $J_3=0.5$ and the transverse field
$h=-0.5$is multicritical where the density shows a slower decay given by
$\tau^{-1/6}$. We also study the effect of ferromagnetic or antiferromagnetic
next nearest neighbor (NNN) interactions on the dynamics of TIM under the same
quenching scheme. In a mean field approximation, the transverse Ising
Hamiltonians with NNN interactions are identical to the three spin Hamiltonian.
The NNN interactions non-trivially modifies the dynamical behavior, for example
an antiferromagnetic NNN interactions results to a larger number of kinks in
the final state in comparison to the case when the NNN interaction is
ferromagnetic.
