There is a major machlokes that tosafos quotes between rabbeinu tam and the r"i regarding the concept of מצוה הבאה בעבירה. Tosafos introduces this by asking a question on u'lah who learns from a pasuk that a korban that is gazul is pasul, from which u'lah derives that yi'ush isn't ko'neh, yet elsewhere u'lah holds that yi'ush is koneh. Rabbeinu Tam holds that yi'ush is ko'neh, but it will not be fit for hakrava in the beis hamikdash due to the problem of מצוה הבאה בעבירה. The R"I disagrees and maintains that if yi'ush qualifies as a kinyan in and of itself, there is no problem of מצוה הבאה בעבירה, because the problem of מצוה הבאה בעבירה only applies when the kinyan is connected to the mitzvah, but if one acquired the object prior to the performance of the mitzvah, it is not a problem of מצוה הבאה בעבירה.
The R"I proves his point that if the acquisition would proceed the mitzvah there would not be a problem of מצוה הבאה בעבירה from the gemara in succah 30a that suggests that a shinuy ma'aseh or shinuy ha'shem to acquire the hadas would circumvent the problem of מצוה הבאה בעבירה. How would Rabbeinu Tam deal with this, since according to him there should be a problem of מצוה הבאה בעבירה even if he acquires it prior to the fulfillment of the mitzvah? The steipler (succah 21) suggests that a shinuy is not merely a kinyan, but rather it is viewed as an entirely new object - פנים חדשות באו לכאן. Therefore, if one was to be ko'neh only through yi'ush (if yi'ush alone worked), that would be nothing more than a kinyan where rabbeinu tam will maintain that there is a problem of מצוה הבאה בעבירה. But when he is koneh in conjunction with a shinuy, it is viewed as an entirely new object and therefore not considered a מצוה הבאה בעבירה. Based on this approach, the halacha in shulchan aruch (649:1) that if one stole a lulav and made a shinuy in it, he can be yotzei his mitzvah, would be even according to rabbeinu tam.