Euonymus alatus (burning bush)escapes via the berries it sets in fall. There are cultivars available for the home landscape that do not set berries, so it's very possible that the mature plants you've seen do not have seedlings next to them. Also, generally the berries are spread by birds and so the seeds are dropped at a distance from the original plant.

So I wouldn't go on a campaign to rip out ALL the euonymus in the US since some of it will not spread.

To answer your question in broader terms, cultivated plants in general escape into the wild in about as many ways as there are for plants to reproduce. Some plants spread by underground runners and can travel amazing distances to do so. Others spread by seed and will use birds, the wind, passing animal fur and other mechanisms to spread. Some can be spread through the dumping of landscape waste; I'm trying to eradicate a patch of crown vetch that grew from the dumping of "dead" plants by someone else.

V.

It is really a complex problem with no simple solution. The main focus has now become identifying new invasives and taking care of them before the problem becomes widespread. I saw a graph of this once. The main ones we know of now (Euonomous, tree of heaven, oriental bittersweet, loosestrife, the honeysuckles, garlic mustard, autumn olive, jap. knotweed, etc.) are too widespread to ever hope of any form of control. Once the scientific community identifies the problem, it is already too late. A woman out of Washigton state has created a computer model that can identify the invasiveness of new plants with about a 80% accuracy rate. This is where the focus of research is going to be placed in the upcomming years.
Don't go ripping out your neighbors shrubs, but if you belong to a garden club you should try and use your club's influence on local nurseries. Hit them in the pocket book by boycotting places that sell known invasives.
I know that I have strayed from your original question. If you go walking in the woods here, there is no question of E. alatus invasiveness. At the garden I work at in the natural area is about 2 acres of just E. alatus. If you do have some E.A. Think about replanting with say, Frothergilla gardenii or F. major 'Mt. Airy" It is a much prettier shrub and is hardy as well.
Sorry to ramble on
tom