Kripke spends a good deal of energy discussing the implications of "transworld identification." He uses this exercise to strengthen his argument for rigid, non-rigid, and accidential designators (Kripke, 1972, p. 48. "Lets call something a rigid designator if in every possible world it designates the same object." I find that Kripkes's evidence gathered in the thought experiment of "other worlds" is extremely gratifying in that it approaches how we think of what is spoken in a much more intuitive manner.
I agree with his assertion that, "We do not begin with worlds, and then ask about criteria of transworld identification; on the contrary, we begin with the objects, which we have, and can identify, in the actual world (Kripke 1972, p. 53)." This seems to me a compelling argument; due to the fact that we usually have a specific object, person, idea, etc. in mind before we determine to suppose properties of that specific item under different circumstances (or in other worlds).
I also think the idea of rigid determiners setting a reference is a much more consistent concept, due to it's lack of reliance on properties as are contextually relevant. This allows us to discuss an object from two very separate, and potentially disparately ignorant perspectives without denying that we are still referencing the same object.
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