Abstract: 
We consider weighted shift operators having the property of moment infinite divisibility; that is, for any $p > 0$, the shift is subnormal when every weight (equivalently, every moment) is raised to the $p$-th power.  By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that they are, under mild conditions, robust under a variety of operations, while also rigid in certain senses.  In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is.  We also consider back-step extensions, subshifts, and completions.