I do not entirely understand the claim that covid-19 vaccines change people's DNA.
As far as I understand, the mRNA from the injections break down fairly quickly and don't do anything to the DNA of the cell. Why would they change DNA and how?

Assuming the vaccines did change the DNA of that individual cell, would this be a problem? Wouldn't the spike proteins that the cell produces because of the vaccine cause it to be destroyed by the immune system? How could the DNA changing of one cell cause all the DNA in someone's body to be compromised?

tl;dr - do vaccines change DNA and how? Would it be a problem if they did?