https://www.sciencedirect.com/search?qs=emissivity&show=25&sortBy=relevance

In the first place, the term thermal inertia is a misnomer. Inertia is generally a property that carries something beyond its equilibrium point, such as with a pendulum. This does not happen with temperatures. Temperatures approach their equilibrium points without overshooting. Regardless of how rapidly a temperature approaches its equilibrium value, it does not overshoot it. If conditions change, the equilibrium temperature may shift, but the system adjusts without inertia.

What most people mean by thermal inertia is the finite time it takes temperatures to adjust to their equilibrium points - that there is a transient in the system response. This transient arises because of finite rates of heat transfer through materials with finite mass. The rate of temperature change in general depends on the heat capacity, the heat flux, and the mass of the material.

The role of emissivity in this response is that it determines the heat flux if radiation dominates or contributes significantly to the total heat flux. The net heat transfer is proportional to the change in the enthalpy.

I don't see any link between the 'thermal inertia' (being the product of mass (or volume) and specific heat capacity, a.k.a. 'thermal mass') and emissivity. Emissivity is a surface property, 'thermal inertia' is related to the bulk material. Example: A roughened copper surface has a different emissivity than a polished surface, but the 'thermal inertia' is the same.