You could try placing all your layers into a smart object, then transform the smart object. You could either put all the layers in the same one, and just turn off the layers you don't need, or you can make a smart object of each layer and alt/opt drag the transform from one to the rest.
Since I will be making revisions to these images and adding more later, the best option for me would be to do something like an alt drag that you mentioned (this is what I do in Illustrator with effects) but I can't seem to do that. What exactly am I alt+dragging?
What exactly am I alt+dragging?
My mistake. I was thinking about filters and alt-dragging them to a new layer. For a transform, you most likely will want to make one master SO, apply the transformation, then make duplicates via copy from the layer flyout menu. Then you can replace the content with the other layers.