What particles can influence the primordial nucleosynthesis?

Equilibrium observables in the Early Universe follow the Markov evolution - the future state of the system is determined only by the current state. The reason for the word "equilibrium" is that any statistical state of the system is uniquely determined by the temperature while non-equilibrium parts of the plasma have memory about the dynamics of the past.

In particular, for the Standard BBN theory we assume that all the relevant constituents of the plasma are equilibrated at the temperatures above few $MeV$ (influence of possible non-equilibrium particles is held to be negligible). This means that any particles existing before, but vanished by the time of BBN, cannot influence its predictions.

The basic analogy to keep in mind comes from the electron-positron annihilation at $T \sim m_e$, that changes the expansion rate of the Universe and the number of relativistic degrees of freedom, but can be mimicked in the later moments by the initial conditions of some different radiation-dominated physics where these particles never existed (leaving the particle physics paradoxes aside).

This creates a constraint on the non-stable non-equilibrium particles that might influence the BBN: their lifetime has to be at least of the same order as the time of decoupling of the first particle species (namely, neutrinos at $T \sim 1 MeV$). As BBN itself is not sensitive to the history of expansion before this moment, it in principle can be freely shifted in time. The particular mapping of the expansion rate and the temperature to some time axis is fixed by the particle physics model being studied.

Rough estimate for the non-equilibrium decaying particles:

$$\begin{equation} \tau \gtrsim 0.01 sec \end{equation}$$

Only models that satisfy this estimate might influence the BBN either by changing the expansion law of the Universe or by adding corrections to the spectra of decoupled species that can affect the equilibrium of specific interactions.