All right, I'm always game for well-executed pedantry. Stand back, all ye who are enraged by pointless rules debates ...
You wouldn't move your opponent's models; if anything, he would move them instead. Page 124 says, "When Reserves arrive, the player picks any one of the units arriving and deploys it, moving it ..."
"The player" is used a number of times throughout the rulebook with reference to one or more models, and it refers not to the player whose turn it is but to the player who controls the related model. Compare the following instances:
Page 22: "Once all the models in a chargrng unit have rnoved, the player can choose another unit and declare another charge if he wishes."
Page 39: "If the Independent Character is within 2" of more than one unit at the end of its Movement phase, the player must declare which unit it is joining."
Page 40: "On a 1-2, the unit comes in frorn the table edge to the left of their controlling player's own table edge; on a 3-4, they come on frorn the right; on a 5-6, the player can choose left or right."
Page 51: "Remember that the player can decide that any model in a firing unit is not going to file its weapon, however, if a model does fi.re, it must do so at full effect (so it cannot fire only two shots from its Heavy 3 weapon)."
Page 85: "When moving a Tank, the player can declare that it is going to attempt to Tank Shock instead of moving normally."
Page 90: "If a unit starts its move outside difficult terrain, the player must declare if he wants his unit to try to enter difficult terrain as part of their move."
Thus, even if a player rolls for all units in reserve on that player's turn, each player would still move his or her own models. While it is true that it is not the opponent's turn, it is not true that page 10 restricts movement of a player's own models to that player's own turn. Rather, page 10 authorizes a player to move his or her models during his or her turn. Page 124 - if one's opponent's models can arrive during one's turn - provides the necessary authorization for the opponent to move the opponent's models even though it is not the opponent's turn.
As for whether page 124 does allow an opponent's models to arrive, I think the question turns on whether we have reason to believe that "each unit" actually means "each unit owned by the player whose turn it is." I can't find any textual reason for that.
So my best pedantic answer: yes, each player rolls for each unit in Reserve, but each such unit arriving is moved by the player owning that unit.