Vertical punches say more power is transferred to the target when using a vertical punch. To be effective, the force of a punch must be transferred from the body into the upper arm, forearm, wrist, hand, and into the target. This force is generated by muscles, but it is transferred through bone. Without bones to act against, the power generated by muscles would be useless. If the bones are misaligned at the joints or the joints are not angled properly, the force transfer will be reduced and injury to the joints may occur.
For the purpose of this discussion, we will consider just the bones in the arm. The single large upper arm bone, the humerus, makes contact at the elbow with two forearm bones, the radius, the larger of the two, and the ulna. The contact the humerous makes with the radius is larger in area, much stronger, and more stable than with the ulna so the radius transfers more force down the arm than the ulna. At the lower end of the arm, the contact the radius makes with the wrist is larger in area, much stronger, and more stable than with the ulna so the radius transfers more force to the wrist than the ulna. When the fist is extended, the radius is naturally in line with the first two knuckles of the hand. The wrist cannot be positioned to make the radius align with the last two knuckles. The Ulna is in line with the middle two knuckles. The only way to line the ulna up with the first two knuckles is to bend the wrist drastically, which is would cause injury to the wrist in a punch. The only way to line the ulna up with the first last knuckles is to bend the wrist, which means the radius is not effectively transferring its greater force.
Vertical punchers say that, as force is transferred down the radius and ulna, a band of very tough connective tissue at the wrist, the interosseous membrane, connects the ulna and radius and causes the forces moving down the two bones to be shared at the wrist. They say that the degree of force transfer via the interosseous membrane depends upon how tight it is, which depends on the amount of twist in the forearm. The tighter the membrane, which they say occurs in a vertically oriented punch, the better the forearm bones share the energy transfer. However, force is transferred through the bones not through connective tissues. All the connective tissues do is hold the bones in position so they make contact. As long as the bones are in contact and stay in contact, the tightness of the interosseous membrane is irrelevant.