I think it depends on what you're practicing, and also who is instructing you in practice.
For example, Vajrayogini in Sakya (Naro Khacho) and Vajrayogini/Varahi in Kamstang, are different.....if one is going to practice one of these, you should choose one, and not try to practice both simultaneously. Maybe taking empowerments in both systems is okay, though with the main yidams, such as this one, there's usually some sort of practice commitment.
Same with Dharmapalas, I think.....some Dharmapalas are more specific to a given lineage, others are more "shared." But even then, the details can differ.....a good example from my personal experience is White Mahakala. The sadhana as practiced by Shangpa (and Karma) Kagyu differs from that practiced in the Sakya, I believe.
The bigger question is why one would want to obtain a given empowerment. If it is with the intent of serious practice, and commitment, I would think you'd want to stick with one lineage at a time.
Some traditions stress the actual Empowerment Ceremony as a path--Nyingma in particular--while others stress that, though it's possible to view that way, empowerment is more of a ripening......for most of us.
These are my observations, so take all this with a grain of salt.
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