It will stain. Yes, it will. *
If it's black, blue, pink, or even a light lavender shade that couldn't possibly do any damage, it will stain. Cotton, synthetic, some unholy mix, or just plain plastic - as soon as you turn your back, it will deposit its surplus dye into the plastic of your poor dolls and then they'll never be able to go shirtless again.
And that's what I learned today!
* Except when it won't! But you won't know until it's too late.