That's accurate enough, but in the early empire at least it was customary for the reigning emperor to adopt someone so that he would succeed him. Which is what the OP meant, I gather.
I have no idea why the tradition didn't persist. It could have something to do with the fact that most European kingdoms (France, Germany and England at least) followed the Sallic law of succession, which originated with the Franks. Which makes me wonder if the Byzantines practiced adoption for the sake of succession.
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