Dry/Moist depends on the vapor pressure of the droplets.. Is sub-super saturated air is entrained into cloud boundaries it will reduce LWC until near drop vapor pressure is in equilibrium with the perturbed cloud environment vapor pressure. As to DSD, this is complex.. I suggest reading

Lasher-trapp, S. G., W. A. Cooper, and A. M. Blyth, 2005: Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Quarterly Journal of the Royal Meteorological Society, 131, 195–220, doi:10.1256/qj.03.199.

Entrainment of dry air into clouds generally leads to a reduction in LWC (evaporation). This is the simple story. As Scott mentions, it gets more complicated when the droplet size distribution is concerned. Among the latest findings: The impact of entrainment on the microstructure of clouds is determined by the extent of "homogeneity" of the mixing, defined by the relative characteristic time scales of turbulent homogenization and droplet evaporation. If turbulent homogenization occurs much faster than evaporation (homogeneous mixing), both droplet size and concentration decrease, but the reduced concentration is only due to change in the volume of the cloudy air. On the opposite end, if turbulent homogenization occurs very slowly (w.r.t. evaporation), droplet sizes decrease but concentration does not change. In intermediate conditions, both droplet size and concentration decrease within the cloudy volume.

For more details: Jarecka et al., 2013: Modeling microphysical effects of entrainment in clouds observed during EUCAARI-IMPACT field campaign. Atmos. Chem. Phys. Discuss., 13, 1489-1526. (http://www.atmos-chem-phys-discuss.net/13/1489/2013/acpd-13-1489-2013.html).

And lets not even start talking about Aerosols and the entrainment of polluted air.. Often people ignore the semi-direct impact here as well..

seed also

question in Researchgate

"In the entrainment zone of stratocumulus at given CAPE, should the size spectrum of droplet formed on aerosols be more broad or more narrow "

If you are interested in the effect of entrainment with the semi-direct effect that have a look the following paper. We found that the entrainment of dry air from above caused evaporation of the smaller droplets in the size distribution:

Hill AA; Dobbie S (2008) The impact of aerosols on non-precipitating marine stratocumulus. II: The semi-direct effect, Q J ROY METEOR SOC, 134, pp.1155-1165. doi: 10.1002/qj.277