Bleeding color usually is a matter of moisture in the frame. Some fabrics also have the possibility of offgassing, which could be a problem in the closed-up atmosphere of the frame. Have you ever walked into an older, smaller, less-well-ventilated fabric store and reacted with watery eyes or stuffy nose, or noticed an acrid smell? The dyes, sizings, and other additives to fabrics are the probable cause.
Tradeoffs, yes. For example, a moderate level of preservation could be achieved by spacing the art/document/object away from the fabric by using an undersized mountboard or some kind of inert, stable spacer. In most cases that would take care of a bleeding color problem.
The offgassing problem, when it exists -- and most of us can't tell when it does -- is another matter. The damage would be done before anyone knows there is a problem. So, prevention is the only safe answer.