How are unmarketable securities like private equity valued upon moving?

it’s not as clear what to do with them since they don’t have a market value at time of BFR change

Assuming you want to bifurcate, For private interests it is proportional to how much time you owned the asset prior to the sale. that has a max lookback period of 10 years. So if you own the private equity for 3 years, move to PR, sell 2 years later, then 2/5 of the gain is PR sourced income. If you hold it while living in PR for 10 years, regardless of how long you owned it before moving to PR, it is 100% PR sourced income and taxed at 5%. There is no importance placed on the value of the asset (eg 409a) at time of your move, unlike publicly traded securities.

The bifurcationh treatment is different for marketable securities

the bifurcation (special eleciton) example from IRS publication 570:

Cheryl acquired a 5% interest in the Alder Partnership on January 1, 2017. On March 1, 2018, when she established bona fide residency in Puerto Rico, her partnership interest was not considered a marketable security. On September 15, 2021, while still a bona fide resident of Puerto Rico, Cheryl sold her interest in Alder Partnership for a $100,000 gain. She had owned the interest for a total of 1,719 days. Cheryl’s territory holding period (from March 1, 2018, through September 15, 2021) is 1,294 days. The portion of her gain attributable to Puerto Rico is $75,276 ($100,000 x (1,294 Puerto Rico days ÷ 1,719 total days)). By reporting $75,276 of her $100,000 gain as Puerto Rico source income on her 2021 Puerto Rico tax return (and the remainder as non-Puerto Rico source income), Cheryl elects to treat $75,276 as Puerto Rico source income.