  We investigate the dynamics of a particle executing a general Continuous Time
Random Walk (CTRW) in three dimensions under the influence of arbitrary
time-varying external fields. Contrary to the general approach in recent works,
our method invokes neither the Fractional Fokker-Planck equation (FFPE) nor the
Stochastic Langevin Equation (SLE). Rather, we use rigorous probability
arguments to derive the general expression for moments of all orders of the
position probability density of the random walker for arbitrary waiting time
density and jump probability density. Closed form expression for the position
probability density is derived for the memoryless condition. For the special
case of CTRW on a one-dimensional lattice with nearest neighbour jumps, our
equations confirm the phenomena of "death of linear response" and
"field-induced dispersion" for sub-diffusion pointed out in [I. M. Sokolov and
J. Klafter, Phys. Rev. Lett. {\bf 97}, 140602 (2006)]. However, our analysis
produces additional terms in the expressions for higher moments, which have
non-trivial consequences. We show that the disappearance of these terms result
from the approximation involved in taking the continuum limit to derive the
generalized Fokker-Planck equation. This establishes the incompleteness of the
FFPE formulation, especially in predicting the higher moments. We also discuss
how different predictions of the model alter if we allow jumps beyond nearest
neighbours and possible circumstances where this becomes relevant.
