This is a great question, and one I would ask before purchasing myself.
First, specimens from the Mini Museum were acquired from a variety of sources. Some specimens I acquired directly from scientists or researchers in the field (ex. Mosasaur, Ankylosaurus). Some specimens were acquired from professional meteorite hunters (Space Gems), reputable collectors (ex. WW II Enigma), and artifact dealers (ex. Samurai Sword). Other specimens I collected on my own or from family heirlooms (ex. San Andreas). Some were also obtained from private land owners (ex. Oldest Earth and Venice). And still others were obtained from open or closed auctions (ex. Steve Jobs Turtleneck, Royal Wedding Cake).
During my nearly four decades of collecting, I've done all that I can to ensure that each sample contained in the Mini Museum is authentic. However, it's important to note that certificates of authenticity are not commonly available from all sources (such as scientists and land owners), but when they are available, I will absolutely share them.
I have also acquired many specimens that would be perfect for a Mini Museum, but will not include them because I am uncomfortable with their provenance. And no matter how much I might personally like these specimens for my own collection, I can't rationalize selling them to others.
I am extremely confident, however, about the provenance for everything included in the Mini Museum. If you're just learning about the Mini Museum, I would also encourage you to have a look at the projects for the First Edition and Second Edition. Dozens of detailed updates spanning the last few years provide a sense of the care with which we've approached the Third Edition.
Finally, I should say that every specimen that I've curated for the Mini Museum is authentic and was acquired ethically and legally. Each mini museum will come with a Certificate of Authenticity signed by me, Hans Fex.