  We present a standard form of master equations (ME) for general
one-dimensional non-Markovian (history-dependent) jump processes, complemented
by an asymptotic solution derived from an expanded system-size approach. The ME
is obtained by developing a general Markovian embedding using a suitable set of
auxiliary field variables. This Markovian embedding uses a Laplace-convolution
operation applied to the velocity trajectory. We introduce an asymptotic method
tailored for this ME standard, generalising the system-size expansion for these
jump processes. Under specific stability conditions tied to a single noise
source, upon coarse-graining, the Generalized Langevin Equation (GLE) emerges
as a universal approximate model for point processes in the weak-coupling
limit. This methodology offers a unified analytical toolset for general
non-Markovian processes, reinforcing the universal applicability of the GLE
founded in microdynamics and the principles of statistical physics.
