This answer seeks to just answer about ambient air entrainment, as per the question (not eruptive plume height, as suggested in the comments) which is a separate question.
According to the book, Modeling Volcanic Processes: The Physics and Mathematics of Volcanism, (Fagents et al, 2012, p. 155-156), your assumption (2nd paragraph) is on the right track, particularly for energetic eruptions (e.g. Plinian).
When the plume initially erupts, it possesses significant momentum, with a greater density and are influenced by pressures considerably greater than the ambient atmosphere. However, once in the atmosphere, the plume undergoes rapid decompression and expands, ascending as a turbulent flow in the convective ascent region (an example is shown below).
Image source: USGS page about the Mt St Helens eruption in 1980 demonstrating the turbulence in an eruptive column.
As the flow continues to ascend, the eruptive column begins to exhibit lateral gradients in temperature, momentum and density, allowing for the mixing with the ambient air at the edges of the column - over time, this mixing continues to the core of the eruptive column by shear mixing. Effectively, 'stirred into the column by the turbulence'.(Fagents et al. 2012; Kaminski et al. 2011).
Image source: MTU.edu
As the entrained air heats, expands and rises, more cooler ambient air is mixed into the turbulent eruptive column (just as long as the eruption continues) - as per the MTU.edu diagram above, the plume rises by buoyant convection.
Entrainment of air into the eruptive column can be increased by the right wind conditions, as observed at the 2011 Kirishima-Shinmoe-dake volcano eruption in Japan (Kozono et al. 2014).
Kozono et al. 2014, Correlation between magma chamber deflation and eruption cloud height during the 2011 Shinmoe-dake eruptions, Earth, Planets and Space
Kaminski et al. 2011, Rise of volcanic plumes to the stratosphere aided by penetrative convection above
large lava flows, Earth and Planetary Science Letters