Q.The dark red rose has invaded 3 of my other roses, like they’ve become part of the other roses but not changing the blooms
The first image is a salmon colored rose that has been invaded by the dark red rose
The second image is a pale pink (peony-style) rose, also invaded by the dark red
The third image is my Cherry Parfait that has been invaded
Thanks for the photos! Your rosebushes are likely grafted roses. I will provide you an article link below about that. There are times when the root stock rose, due to its hardiness, sends up its own shoots that then turn into invasive canes. I mean Very Invasive too! What you have to do is to follow each such cane down into the ground an inch or so, then prune it out. Do this for all of the invasive canes. This will discourage the root stock from sending up more. You will need to keep close eye on things though and prune out such shoots as soon as you see them. These shoots usually come up from below ground level and are not the top desired part of the rosebush. If left to grow they will indeed take over and will rob the top and desired part of the rosebushes affected of vital nutrients. Thus killing the upper or top part of the grafted rosebush.
Article Link: https://www.gardeningknowhow.com/ornamental/flowers/roses/own-root-roses-grafted-roses.htm