“What are the consequences and/or benefits of mixing synthetic and non-synthetic oil in an engine crankcase, or any other application such as a gearbox? I have been told that a crankcase can be drained of non-synthetic oil and synthetic oil can be added without flushing any of the prior oil residue."

In general it is never a good idea to mix two different oils in any application. If this is unavoidable, there are two major concerns, specifically incompatibility between the base oil, and incompatibility between the additives. The overwhelming majority of synthetic oils, particularly crankcase oils are PAO based. 

From a base oil standpoint, there are usually no problems with mixing a PAO synthetic and a mineral oil since both are hydrocarbon based and chemically very similar. However, extreme caution should be used if the synthetic oil is non-hydrocarbon based.

As for additive incompatibility, the biggest danger is mixing oils with very different additive packages, particularly mixing those with acidic additives, such as most AW and EP oils, with oils containing alkaline additives commonly used in crankcase oils. In this case, mixing can result in a number of problems including additive drop out, loss of demulsibility, foaming problems and reduced oxidation stability.

The best approach is to select the appropriate lubricant for the application at hand and stick with it!  If you are forced to mix oil of different types, it is important that either the lubricant manufacturer is consulted and/or compatibility testing is performed to avoid potential problems.