Not if the dye is set properly with heat.
The citric acid in the KA is what makes it work. If one is using food coloring, one must add an acid, like vinegar, in order for the dye to work.
Bleeding is caused because the fiber can't take up any more dye and the remaining unattached dye molecules just cling to the outside of the fiber. When water is put on them, they flow off or bleed. This will happen with any dyepot that has more molecules of dye than the fiber can absorb. Or the dye wasn't heat set properly.
I've used the microwave, a pot of simmering water, the crock pot, plastic bags in the sun, :hot: gallon jars in the sun, and the oven as heat. All have worked and I've not had any bleeding at all.
KA and food coloring, as I said, are environmentally sound dyes. All "real" dyes have mordants that can be dangerous to children and other living things. Some of them should be treated as hazardous wastes and not poured into the public sewer systems.
Play with some KA and see how much fun it is. A search will bring up tons of KA tutorials.