Unless there are yellowish pustules on the underside of the leaves of  your tayberry and marionberry plants it looks like you have Septoria leaf spot, rather than cane and leaf rust.  Septoria leaf spot is caused by a fungus, Sphaerulina westendorpii - (it used to be called Septoria rubi).  Septoria affects trailing berries, which is why your raspberry is not affected.  Septoria causes small purplish leaf spots that turn brown with age.  They form white areas in the middle of the spots (visible in your pictures), with minute black fruiting bodies (pycnidia) which produce the infective spores.  Cane and leaf rust form similar-looking spots on the leaves, but have characteristic yellow pustules on the underside of the leaves.

Sephoria spreads with rains.  To control sephoria cut and remove the old fruiting canes (floricanes) after harvest to remove the infected leaves and canes.  Trellis the new (primocanes) canes in August through early September - get them off the ground and onto trellises before the fall rains start.  Control weeds as they provide good cover for the fungus.  Promote good air circulation so the plants will dry out.  Using drip or soaker hose irrigation helps keep the leaves dry.  Next spring watch your blackberry vines for early leaf spots (~0.12 mm and purple).  If there are too many you can spray with products labeled for home use (noted by an 'H' in a rectangle in this article) from OSU  which gives additional information and pictures, Blackberry (Rubus spp.) - Septoria Leaf Spot  https://pnwhandbooks.org/plantdisease/host-disease/blackberry-rubus-spp-septoria-leaf-spot.  Be sure an follow all directions for use and safety on the label of any products used.