If the most recent tiles were newly installed then they won't contain asbestos.
I am now wondering if maybe the new tiles are cheapo self-stick tiles that are so popular today. If so, they should peel right up with a little encouragement.
Have you tried to remove any of them?
If they were installed using the proper adhesive then they are installed with a contact adhesive and they won't be so willing to let go. You can use a heat gun but it is a slow slow process usually.
The first thing to do is to remove a few and see what you are dealing with.
The new adhesive will likely contain the old tiles below and scraping on them a little shouldn't be an issue of exposing any asbestos.