My "high school understanding" comment was poorly-written, and I was really trying to express that, at the end of the day, DNA is DNA, and our bodies can't tell the difference when we consume it. Different genes could indeed cause unexpected behaviors or proteins to be expressed in plants, but these are easy to spot during the testing that occurs, and largely, the issues that pop up during this process are failure to get cellular material to accept a transgene, and then to regenerate a whole plant after modified cells have the desired transgene.
Put quite simply, the engineered varieties undergo toxicity studies and in-depth chemical and molecular analysis, and their agronomic impact is assessed through things like soil samples, and all of this data is compared to the same studies which are also done on their non-GM counterparts to assure that nothing unexpected has changed, and that the end result is "substantially equivalent." It usually costs around $30m, and takes between 10 and 15 years to complete the testing, regulatory, and approval process over many many generations of the engineered variety.
I'm really not aware of any studies like what you're looking for, but the data would be much more valuable looking at specific varieties of organisms, rather than based on what method was used to create them. Even then, confirming their chemical, molecular, nutritional, and agronomic equivalence with their non-GM counterparts really negates the need for those sorts of studies, because the crop ends up being indistinguishable (aside from a few crops engineered for the end consumer, such as vegetables with higher levels of antioxidants or higher levels of certain vitamins).
A wall of text follows.
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I'm not really familiar with the methods used for varieties derived from breeding or mutagenesis, but I recall the last time that I looked, there was not any major standard testing being done there.
The first step in adding an isolated gene to a genome is to do a few modifications to the genetic payload itself. A marker gene is added as well as a promoter, and a termination sequence is appended to the transgene. The marker gene is used to differentiate which cells have successfully taken the transgene. In short, the targeted cells are grown in a medium that is normally toxic to the target plant. The marker gene encodes proteins which provide resistance to that medium, thus, the cells that survive in the medium are the ones which have had successful uptake of the gene. The promoter sequence controls when and how the transgene is expressed over the life of the plant, and the termination sequence simply marks the end of the payload.
Once you have your genetic payload, there are several methods used to get it into a plant cell. The first, and simplest, is a gene gun, which quite literally blasts a cluster of cells with particles coated in genetic material. Some of the cells survive and successfully envelope a particle coated with DNA, which eventually integrates into a chromosome. This is used for plants which don't readily accept other forms of modification, and was the method used to create Bt corn as well as golden rice.
Agrobacterium is another method which is generally more precise about delivering genetic material to a desired location, but various plants are resistant to the method. Unmodified, the agrobacterium infects plant cells through open wounds, and deposits a DNA payload during replication which causes tumors. The tumor-causing DNA payload is removed, replaced with the transgenic payload, and the bacteria carries out its job, infecting the plant and depositing the payload for a desired trait.
Soon, we will probably start seeing newer methods such as CRISPR/Cas9 become more popular (assuming they aren't as resisted as agrobacterium can be).
By precisely controlling the genetic payloads added to the genome, it is far easier to monitor protein expression and see exactly what has changed throughout a plant's lifecycle.
If and when successful regeneration of a whole plant occurs, evaluation of their offspring occurs. The first is a standard 90-day animal-feeding toxicity study. Data from this study is also compared to data from animal studies of the crop's non-GM equivalent. Various studies are also done on the source of the transgene. Gene characterization is also done to examine a gene's molecular structure and sequence, and look at its relationship to similar genes and their exhibited effects.
Compositional analysis and comparison is also done between the GM and non-GM varieties. The varieties are grown adjacent to each other in various locations and are then observed. Various plant components are analyzed and compared between the GM and non-GM varieties using GC/MS and other methods to examine nutrients, vitamins, proteins, fat content, fiber content, and so forth. This is to ensure that the new crop is chemically and nutritionally equivalent to its non-GM counterparts.
Beyond the compositional analysis, environmental/agronomic impact and equivalence are also assessed between the GM and non-GM varieties, usually during the period they're grown leading to compositional analysis. Soil samples are taken to examine nutrient absorption, and things like root structures are also examined and compared.
At this point, assuming all has gone well, a genetically-engineered crop is considered to be "substantially equivalent" to its non-GM counterpart.
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I hope some of this was helpful, and that it made sense and that I wasn't too long-winded.