Leather is made by tanning it, dying it, then putting a finish on it. If the finish isn't quite good enough, some of the dye may rub off. There are different kinds of finishes;
some quite nice leather may still leak dye, even though it's something you're more likely to associate with poor quality leather.
If I were concerned about dye leakage from a leather item with a smooth or especially a glossy finish*, here's what I'd do:
On a small hidden area of the item, test the leather surface with glycerin saddle soap (I like sprayable liquid stuff for the convenience). Dab a little of the soap in that spot. If it discolors, stop; the leather won't tolerate messing with it. If it doesn't, go ahead and wipe the whole bag with soap, rinsing your applicator from time to time to get rid of any dye it's picked up. Make sure to get in the cracks, seams, edges, etc. You can use paper towels, rags, or small sponges.
This should help get rid of any dye that has already come off the leather itself and is just on the surface.
Let it dry completely.
Now you'll want to make doubly sure that what dye is in the leather stays in the leather, and doesn't rub off. You can put a leather finish called Carnauba Cream over just about any other type of finish, and it's a very good finish for flexible or softer leathers (as compared to stiff ones like, for example, shoes).
Again, test on a small hidden area first. Then you can wipe the whole item down with a thin layer of the Carnauba Cream, making sure to get edges, cracks, etc. I prefer to use wool for this, but a very soft cloth, like the remains of a well-worn 100% cotton T-shirt or flannel, should be suitable. Once it dries completely you should have an additional layer of protective finish between the dye and your clothes

Test again, as you have already done, with a damp white cloth to see if any red comes off. If so, you can try another layer of Carnauba Cream.
Carnauba Cream is also a conditioner, and is good for leather, so don't worry about damaging the item by applying something on top of the nice leather surface you already have. It's even good for applying to items you already have that are starting to look a bit lackluster (after a similar cleaning, as above).
* If your leather has a nubuck or sueded surface, you don't want to do any of this.