There are many good responses here already and I’m going to add a perspective I don’t see.
As a coworker: don’t do anything. Sometimes work is an opportunity for someone to escape the personal tragedy that has swept through their life. It’s a chance to think about something else for a while, something grounded and neutral. A time to feel normal.
That can be a very helpful support. Reminding them of the personal stuff may ruin that.
I know that you are a nice human and want to be good to them, but sometimes we need to realize that we are just co-workers to someone and not friends and family. Knowing your place can be an incredibly respectful and positive thing to do.
Add this to your other ideas about how to show sympathy. It’s just one angle. I’m not saying this is always right 100% for everyone and bar nothing.
It’s not the only way. We’ve been relying on immigration to do this for us for a long time. American politics is struggling with the tension between racist idiots who don’t understand that immigrants are crucial to our economy, and those who do understand it. You can not be racist AND not understand it and that is fine too, no problem.