Then make sure that the file extension is the same for the new site. So if the old site's extension is html then change the home page of the new site to index.html and all pages will export as .html

The can be browser issues if someone goes to page.htm and the actual file name is page.html

And if a browser finds two pages, index.htm and index.html (and this may have changed, I'm not sure) even if the index.htm is newer, the browser will load the index.html file.