The rate at which atmosphere is entrained and mixed into relatively dense explosive volcanic jets of gas and pyroclasts (ash, lapilli) determines whether they rise into the atmosphere as buoyant plumes or collapse to form pyroclastic flows. Here, we use analog experiments on grid-stirred turbulence to isolate particle inertial effects related to the motions of ash and lapilli on the quantitative dynamical conditions for the onset of turbulent entrainment and mixing. From energetic considerations, we estimate the distinct effects of ash and lapilli on the entrainment rate. We find that, whereas large ash- and lapilli-sized pyroclasts contribute angular momentum to entraining eddies and may enhance entrainment into volcanic jets by 37% compared to expectations without inertial particles, ash is dissipative and may reduce entrainment by 14%. We also show that the internal turbulent mixing properties are largely insensitive to particle inertial effects. Our results predict that highly explosive events involving water or ice should, for example, feed plumes that rise higher than eruption columns that are enriched in larger pyroclasts, although these events are also more likely to produce pyroclastic flows.