I have been moving away from selling fresh grafted material to avoid the higher risk associated with new grafts.  Often propagators will remove understocks prior to shipping to a grower, which can often be devastating to a buyer, resulting in low vitality rates. For this reason, young grafted plants that I ship may not have understocks cut back entirely when you receive them. It’s best to reduce understock slowly and remove entirely when understock is less significant in relation to caliper of new plant.  

In caring for young pines or any pine for that matter fungicides are essential.   A couple applications of liquid copper fungicide in spring months followed by one summer and fall application is very beneficial in preventing needle blight and greatly reduces the fatality rate of young pines and adventitious growth.  
Black Pine Understock
Japanese black pine under stock has long been the standard for many pine trees used in bonsai art.  The Black pine is less susceptible to heat stress and diseases inherit with container culture.  Granted, many enthusiasts have been able to air-layer a tree or perhaps even succeeded on rooting a pine cutting. Yet, they will need to grow tree for years avoiding freezing temperatures to be considered hardy enough to survive on plants own roots.   In the case of white pine, most often the root collar or nebrari of a black pine can more favorable depending on the development of low graft. 
I have been making significant headway on improving understock selection, using quality Japanese black Mikawa seedlings and trying to be innovative while improving quality of my grafts. If you have any questions regarding culture of your grafted plants or your next purchase please don't hesitate to contact me.