  We consider a one-dimensional classical ferromagnetic Ising model when it is
quenched from a low temperature to zero temperature in finite time using
Glauber or Kawasaki dynamics. Most of the previous work on finite-time quenches
assume that the system is initially in equilibrium and focus on the excess
defect density at the end of the quench which decays algebraically in quench
time with Kibble-Zurek exponent. Here we are interested in understanding the
conditions under which the Kibble-Zurek scalings do not hold and in elucidating
the full dynamics of the defect density. We find that depending on the initial
conditions and quench time, the dynamics of the defect density can be
characterized by coarsening and/or the standard finite-time quench dynamics
involving adiabatic evolution and Kibble-Zurek dynamics; the time scales for
crossover between these dynamical phases are determined by coarsening time and
stationary state relaxation time. As a consequence, the defect density at the
end of the quench is either a constant or decays following coarsening laws or
Kibble-Zurek scaling. For the Glauber chain, we formulate a low temperature
scaling theory and find exact expressions for the final defect density for
various initial conditions. For the Kawasaki chain where the dynamic exponents
for coarsening and stationary state dynamics are different, we verify the above
findings numerically and also examine the effect of unequal dynamic exponents.
