Relatively few long-term experiments have been done surveying invasive plants in the same place, so we know relatively little about the long-term dynamics of invasive plants. A study on Canada thistle, Cirsium arvense, on burned sites in Yellowstone reveal that over seven years thistle has disappeared from some sites as native vegetation has recovered and at other sites its relative cover has declined. Where it has remained or newly appeared, sites tend to have more fertile soils and areas with bare ground.
Wright, B. R. and D. B. Tinker. 2012. Canada thistle (Cirsium arvense (L.) Scop.) dynamics in young, postfire forests in Yellowstone National Park, northwestern Wyoming. Plant Ecology DOI: 10.
The article encourages the use of weed risk assessments, but points out that with new suppliers, a history of invasion by a particular plant might be hard to come by and invasion history is an important factor in determining risk. On a positive note, the authors point out that one nursery promoting drought tolerant plants has mostly increased the number of drought tolerant native plants it offers.