Do the ninken let people carry them around? Pakkun probably has to because he's the littlest, right?
Pakkun has only certain people he will 100% willingly let carry him around. Kakashi (of course), Gai (a little begrudgingly but if Kakashi's not available this is a good second option) and Tenzo (he likes Tenzo, but if its between Gai and Tenzo, kakashi's boyfriend wins by default).
If all of the nindog's are out, Bull is Pakkun's go to ride. He can still move fast on Bull and he's right there in front of the pack for anything.
If, however, none of the above options are available, he will simply go with whoever hasn't annoyed him yet. It's a process of elimination and it's kind of funny to see how some of them react when he sits on their shoulder.
As for the other dogs, most of them don't usually need to be picked up but Kakahsi and Gai will absolutly do it anyways. Gai has been seen carrying an injured Bull after a mission while Kakashi was carrying Akino because Kakashi wouldn't let them disappear with such big wounds and demmands that someone heal them (he is super protective of his ninkin)
Akino however does LOVE to be picked up. He will 100% jump into Kakakashi, Gai or Tenzo's arms because he likes all three of them and it's fun to see how they react to the 'carry me' jump.
Bull is the only one who has never asked to be picked up, but that does not mean he has been saved from it. Gai has given that particular ninkin piggyback rides simply for being awsome (gai adores bull. They're like bff's when the whole gang is out. Kakashi actually worried some days that gai might steal bull)