The problem isn't with GMOs themselves, its with the company(ies) that produces them. For example, if you don't use Monsanto soybeans and recycle your beans - Monsanto will sue you saying that you are breaking a contract with them by reusing their seeds. It's then up to the small farmers and recyclers to prove in court that they aren't.
This is indeed an issue, however there have been many animal studies done that show how the inserted DNA can split off and recombine with your DNA or that if your gut microbes. I believe this is what ASA is talking about.
They are also experimenting with RNA based modifications in Australia. There is speculation that this form of GE will jump easier than DNA based. They have also supposedly found plant crops contaminated with viral RNA that would make the consumer extremely sick/die. Because it would attack via RNA it would attack your body like an autoimmune disease.
