Continuing this discussion.
The gemara says that Rav Mari was punished with the death of his animals. Rabbeinu Gershom explains that he wrongdoing was that he sold his animals through his adult children who were purchased them on behalf of the goy [זכייה]. People who saw this were liable to misunderstand and would sell them through their minor children [קטנים] which doesn't work and the animals would remain in the property of the Jew and be obligated in Bechora. The Jew wouldn't know this and treat them like regular chullin animals and come to sin.
This is very difficult to understand [if you think about it. Otherwise it is easy to understand:-)]. We know that according to a whole SLEW of rishonim זכייה is מטעם שליחות. The notion of acquiring on behalf of someone else without their knowledge is a type of agency [because we know that they want it since it is to their advantage]. Now, there is no שליחות for a goy so how is there זכייה on his behalf.
As they say in Lakewood "I pashut don't get it....."
Rav Wahrman [see linked post above] points us to a Tosfos in Temurah [2b, I KNOW you were thinking about that one....:-)] that explains the teaching of R' Yehuda that a goy can make temurah ["exchange" the kedusha from one animal to another] to be talking about a case where the goy said anybody who wants to do temurah on my animal may do so. The acharonim wonder about this explanation. There is a rule אין שליחות לעכו"ם? So how can a Jew be his agent?? It doesn't work.
The answer is found in a Ran in Nedarim [36b] who says that sometimes שליחות is not necessary and a גילוי דעת - declaration of intent, a granting of permission, is enough. Therefore, if I say that anyone may come and separate terumah from my fruits and a person who is מודר הנאה [from whom I am forbidden to receive benefit] comes and separates, it is not considered that I benefited from him. He is not my שליח because all I did was declare my intent and give permission but didn't explicitly make him my שליח.
With this principal in hand we can answer the question of Rav Eliezer Silver who asked how the gemara in Temurah [9a] says that one can perform temurah on the animal of someone else when the owner declares כל הרוצה להמיר יבוא וימיר - Anybody who wants may come and do temurah. The Yerushalmi [Kiddushin 2/1] says that there is no concept of שליחות with respect to Temurah?! So how does anyone have the mandate to do temurah for someone else?
Not only that, but the very act of Temurah is sinful and אין שליח לדבר עבירה, so how can someone be my shaliach for Temurah?
But now we understand. When someone says כל הרוצה להמר יבוא וימיר it is NOT considered שליחות just a גילוי דעת.
Now Rabbeinu Gershom is sweet. He could sell them to the goy through his children with the mechanism of זכייה and there is no problem of אין זכייה לגוי. Why? Since an animal of hefker is פטור from the בכורה [as we saw in the previous post] all the Jew has to do is remove his rights to the animal and passively allow the goy to take control of it. This is not considered שליחות but a mere גילוי דעת that the goy has permission to acquire the animals. It is NOT that the father APPOINTED his children שליחים on behalf of the goy [that wouldn't fly] but that he was מגלה דעת that he is no longer the owner and automatically the goy can take them over.