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Averaging correlation matrices based on different models, but the same data, is commonly done. If the correlation matrices are derived from return series, is it proper/common to also average the correlation matrices based on return series of the same underlying price series, but different periodicity?

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I don't know how common this is, but I've seen it done. Many risk model vendors (Northfield, Axioma) allow the blending of different risk models with different periodicity (e.g. a shorter horizon risk model blended with a longer horizon risk model). Here's a Northfield deck about this:

https://www.northinfo.com/documents/779.pdf

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