In a circular pattern only the original part instance can be moved if it is unconstrained. All copies generated by the pattern are fully constrained by the pattern and cannot be moved. If the original part instance is moved then the copies will move in a relative fashion.
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Addendum: In a rectangular pattern, if no parts are constrained, then the entire pattern may be moved as a unit by grabbing any part. The relative positions of the parts within the pattern will remain fixed by the parameters of the pattern command.

A rectangular or circular pattern can have constraints applied to any of it's parts. The constraints do not have to be on the original part(s). Constraints may be added to both fix the position of the entire pattern and to attach more components to the parts in the pattern. However, I would avoid this practice if possible. It can be difficult to find and edit constraints embedded within patterns. Subsequent edits to patterns can have unexpected results including constraints being deleted without any indication. I find it best to only apply constraints to the original part(s) in a pattern before the pattern is created.