Experiments of inserting foreign genes into yeasts have been successful but yeasts do not have much non-coding DNA to bother with. The insertion of foreign genetic material into mice though is quite complicated:
Using recombinant DNA methods, build molecules of DNA containing
* the structural gene you desire (e.g., the insulin gene)
* vector DNA to enable the molecules to be inserted into host DNA molecules
* promoter and enhancer sequences to enable the gene to be expressed by host cells
From: Transgenic Animals
Not so easy and note that they have to take not just the gene, but other sequences from the genome in order to make it usable.
With current protocols for the creation of transgenic mice by embryo microinjection, the site of integration is not predetermined, and, for all practical purposes, should be considered random. Microinjection allows one to add, but not subtract genetic material in a directed manner; if a particular experiment leads to the insertion of an novel version of a mouse gene into the genome, this novel allele will be present in addition to the normal diploid pair. Consequently, only dominant, or co-dominant, forms of phenotypic expression will be detectable from the transgene....
... recessive phenotypes are most likely due to the disruption of some normal vital gene. In less frequent cases, a transgene may land at a site that is flanked by an endogenous enhancer which can stimulate gene activity at inappropriate stages or tissues. This can lead to the expression of dominant phenotypes that are not strictly a result of the transgene itself. 39 For all of these reasons, it is critical to analyze data from three or more founder lines with the same transgene construct before reaching conclusions concerning the effect, or lack thereof, on the mouse phenotype....
Unless a particular transgenic insertion causes an easily detectable, dominant phenotype, the presence of the transgene in an animal is most readily determined through DNA analysis....
In the vast majority of cases, however, homozygous Tg/Tg animals will be indistinguishable in phenotype from heterozygous Tg/+ animals, and without a recessive phenotype, the identification of homozygous animals will not be straightforward.
From: Transgenic Mouse
Note that the above shows the insertion of an additional gene. Note that the random insertion can cause death if is ends up in a vital part of the genome. Note that the inserted gene must be a dominant one. Note that even if all the above happens though - that it is hard to figure out if the gene is present in subsequent generations because most likely being an additional gene it does not get used! They have to basically search the genome to see if the gene was passed on!