Have your Pakistani trees ever produced fruit? Those catkins maybe the fruit, which is trying to set on the 2 trees, but dropping due to lack of irrigation. They may also benefit from some fertilizer if you haven't added that already. A photo will help ID if they are male or female flowers when you get the chance.
I'm near Richmond, SW Houston, and am growing a few Pakistani's, and 6 other mulberry varieties. Mulberries are drought tolerant trees, but while they are fruiting they do benefit from regular irrigation.
Sounds good! Keep an eye on the trees, and you can always graft onto the non-fruiting branches in the future if you like.