Cost basis when you transfer shares

We recently did a gift transfer to a family trust and assumed the cost basis from the original certificate would just carry over, but after the transfer completed I wasn't totally sure whether it had or whether we needed to go back and manually update it. I ended up checking the new certificate and it looked right, but I honestly couldn't tell if that was by design or luck.

Now we have a couple of transfers coming up and I'm realizing I don't actually understand the rule. Does Carta use the original basis? Calculate a new one from the cash paid? Something else entirely? And does the answer change depending on the transfer reason you select?

Would love to know if anyone has mapped this out, or if there's documentation I'm clearly missing.