There are many papers that seek to constrain potential dark matter theories based upon Big Bang Nucleosynthesis. Also, the Lithium problem (a deficit of Lithium-7 from what would be expected from BBN) is itself a problem for the standard LambdaCDM cosmology (but could be due to something as simple as a lack of understanding of how post-BBN star chemistry impacts Lithium-7 levels observed today).
This powerfully constrains, for example, the cross section of interaction that a dark matter candidate can have with ordinary matter.
For example, it rules out the shaded area of parameter space (the shaded area on the left) for a particular class of dark matter candidates in the chart from
https://arxiv.org/abs/2107.12377 because it would make it appear that there are more effective neutrino species that astronomy observations support for low mass dark matter candidates with meaningful (i.e. at least as strong as neutrino) interactions with ordinary matter.
This limitation on turns out to be quite useful, because light dark matter candidates are particularly hard for direct dark matter detection experiments to exclude because the noise from neutrinos themselves can overwhelm a dark matter signal in those experimental designs. Where direct detection experiments are available they usually provide the strongest exclusions of the dark matter candidate parameter space, while BBN provides an almost completely non-overlapping exclusion for lighter dark matter candidates with comparable cross-section of interaction cutoffs.
Together these constraints largely rule out dark matter candidates that interact with ordinary matter at a strength equal to or greater than neutrino interactions with ordinary matter via the weak force, across almost the entire range of dark matter candidate masses for a very large and generic class of dark matter candidates.
For example, this is very constraining for supersymmetric particle candidates, since these candidates generally have weak force interaction strengths that are well defined, a priori, in the theory from which they are derived.
Another recent paper derives a similar constraint for essentially the same reasons (the paper with the chart is mostly looking at another issue and incorporating prior BBN constraint on DM research).
https://arxiv.org/abs/2106.07122
BBN can also be used to constrain "sterile neutrinos" (a subset of "heavy neutral leptons") that oscillate with ordinary neutrinos but otherwise don't interact with ordinary matter.
https://arxiv.org/abs/2006.07387
BBN also constrains DM that decays to visible particles.
https://arxiv.org/abs/1910.06272
Macroscopic dark matter candidates, meanwhile, can be constrained by the change in the baryon density between the end of the Big Bang Nucleosynthesis (BBN) and the Cosmic Microwave Background (CMB) decoupling, inferred from astronomy observations.
https://arxiv.org/abs/2105.13932 Similarly, it imposes constraints on primordial black hole dark matter theories.
https://arxiv.org/abs/2002.12778
BBN has also been used to constrain non-LambdaCDM cosmology theories.
https://arxiv.org/abs/2104.11296
In general, however, such constraints are tricky because there are usually multiple different mechanisms that can have the same BBN impact (or lack thereof). For example, a MOND inspired cosmology without dark matter doesn't have the "Lithium problem" found in LambdaCDM BBN fits to astronomy observations, but has a lower statistical significance deuterium problem.
The lithium problem can also be addressed with light dark matter candidates produced partially or in full, non-thermally.
https://arxiv.org/abs/1912.05563
As a general rule, the more sterile a dark matter candidate is in terms of interactions with ordinary matter, and the more symmetrically (more precisely, the more isotropically and homogeneously) dark matter is distributed at the time of BBN, the less of an impact it has on BBN. And, because BBN happens so early after the Big Bang (between
about 10 seconds and 20 minutes after the Big Bang), there isn't a lot of time for anisotropy or inhomogeneity to emerge in dark matter particle distributions from random quantum variations by then.