I don't think mRNA is typically reverse transcribed into DNA and integrated into the host genome. However, the effects of the mRNA "jab" could be quite bad, imo. The body cells expressing the "spike" protein will provoke a host response that could ignite auto-immunity plus the antibodies produced are likely to be highly specific, perhaps driving immune escape by the pathogen or else contributing to antibody dependent enhancement of disease.
I do not rule out some possibility of host genome modification, but it seems less of an issue that other problems with this approach. There is an enzyme called reverse transcriptase which could allow RNA to be transcribed into DNA.
Personally I suspect the risk of genome modification by the mRNA jab is less than that of the adenovirus vector vaccines. Plus, many of our current vaccines contain small fragment human DNA from abortions. A much larger issue of genetic modification is possible with these jabs, I think.