  We investigate the possible impact of diffusion on the abundance of helium
and other primordial elements during formation of the first structures in the
early Universe. We consider the primary collapse of a perturbation and
subsequent accretion of matter onto the virialized halo, restricting our
consideration to halos with masses considerably above the Jeans limit. We find
that diffusion in the cold and nearly neutral primordial gas at the end of the
Dark Ages could raise the abundance of primordial elements relative to hydrogen
in the first virialized halos: helium enrichment could reach $\delta Y_p/Y_p
\sim 10^{-4}$ in the first star-forming minihalos of $ \sim 10^5-10^6
M_{\odot}$. A moderate (to ~ 100 K) preheating of the primordial gas at the
beginning of cosmic reionization could increase this effect to $\delta Y_p/Y_p
\sim 3\times 10^{-4}$ for $\sim 10^6 M_{\odot}$ halos. Even stronger abundance
enhancements, $\delta Y_p/Y_p$ ~ a few $10^{-3}$, may arise at much later,
post-reionization epochs, z ~ 2, in protogroups of galaxies ($\sim 10^{13}
M_{\odot}$) as a result of accretion of warm-hot intergalactic medium with T ~
10^6 K. The diffusion-induced abundance changes discussed here are small but
comparable to the already achieved ~ 0.1 % precision of cosmological
predictions of the primordial He abundance. If direct helium abundance
measurements (in particular, in low-metallicity HII regions in dwarf galaxies)
achieve the same level of precision in the future, their comparison with the
BBN predictions may require consideration of the effects discussed here.
