Rules have been that players traded across leagues in the offseason can not be kept in either league
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Players traded into into a league during the offseason are not eligible to be kept in either league. In the Trout example above, if that trade took place in December, he is no longer in the AL and must be dropped from team who owned him in the AL. In the NL he is not eligible to be added since there are no waivers/FA adds during the offseason. So, he would be eligible to be drafted in the next NL draft, but no team would be able to acquire him on their NL roster before then.
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And yes players signing into a new league as a free agent can not be kept in either league