I can't tell you about any specific diamond-tracking company, but I can address the complications with tracking any physical good on a blockchain.
In order to track a physical object on a blockchain, that physical object needs to be able to reduce to a small piece of data, such as a hash digest. With digital goods, this is easily accomplished with a cryptographic hash function, but physical objects present some difficulties. You'd need a method of reliably and repeatably coming up with the same reduced value from the same physical object. If a diamond were "hashed" in London today by one jeweler, you would need to be able to expect that if it were hashed again by a different jeweler in New York in 20 years, the result would be the same. Not just close, but exactly the same. This is probably where the physical "passport" you speak of fits in, but in that case, you are relying on the chain of custody of the passport. The blockchain would be tracking the passport, not the diamond. How the passport is identified to belong to a certain diamond is something you'd have to ask a jeweler.
You would also need to trust whatever layer is built on top of the blockchain, and that they are following the rules they claim to be following. In this case, it would be whomever issues these passports. For example, what would happen if the passport-issuing authority issues two passports for the same diamond? The passports could be traded separately on the blockchain, but there is still only a single physical good...but with two blockchain-legitimate claims to it. Also, who is in charge of enforcing the ownership of the physical good? If I have the diamond, but you have the blockchain token, who is going to physically take the diamond from me and give it to you?
It might be wise to pose these questions to whichever company is claiming to be able to track ownership of a physical good by using a blockchain.