  We extend our initial study of diffusive nuclear burning (DNB) for neutron
stars (NSs) with Hydrogen atmospheres and an underlying layer of proton
capturing nuclei. Our initial study showed that DNB can alter the photospheric
abundance of Hydrogen on surprisingly short timescales ($10^{2-4}\yrs$).
Significant composition evolution impacts the radiated thermal spectrum from
the NS as well as its overall cooling rate. In this paper, we consider the case
when the rate limiting step for the H consumption is diffusion to the burning
layer, rather than the local nuclear timescale. This is relevant for NSs with
surface temperatures in excess of $10^6 {\rm K}$, such as young ($<10^5$ yr)
radio pulsars and accreting NSs in quiescence. When downward diffusion is the
limiting rate in DNB, the rate of H consumption is suppressed by 1-2 orders of
magnitude compared to a DNB estimate that assumes diffusive equilibrium. In
order to apply our ongoing study to young neutron stars, we also include the
important effects of strong magnetic fields ($B \sim 10^{12} {\rm G}$). In this
initial study of magnetic modifications to DNB, we find that the H burning time
is lengthened by 2-3 orders of magnitude for a $10^{12} {\rm G}$ field.
However, even for NSs with dipole field strengths of $10^{12}$ G, we find that
all of the H can be burned before the pulsar reaches an age of $\sim 10^5 \
{\rm yr}$, thus potentially revealing the underlying proton-capturing elements.
Finally, we conclude by providing an overview of what can be learned about
fallback and pulsar winds from measuring the surface composition of a young NS.
