Tolley All good questions, see below.
What happens to the bones?
The bulk of the skeleton is dust, but larger bones tend to stay in shape/ tact. The lower half of the skull remained, although it is all white and charred and very delicate. They see how many bones there are and how many mourners and break them up to ensure that all family members are able to place a piece of bone into the urn. This is done by the chief burner bloke, he is dressed in a simple uniform, the sort you could see down the fish market or something, no robes. The bones are broken with two large chopsticks.
You and your missus put bits of rib bone into an urn - presumably to be kept by the family, while your daughter has put a piece of bone into a box she is holding?
Yes, it seems that the main urn is used for the placement of the bits of bone by the family, and a chosen piece of bone is placed into a box as a special memento. i haven't got to the bottom of what or why this is, other than the throat bone was chosen yesterday as being in perfect (white charred) form. I will need to report back on ow this special piece is selected.
Does she keep this as a memento?
It is kept at the side of the main urn on the family altar. Most Japanese homes have some form of small family altar for their dead. It is called a Butsudan, and can be simpple or ornate and expensive. Both will be at the main faimily home.
https://www.tofugu.com/japan/butsudan/
Do they grind the bones down into ash in a cremulator like in the UK, or are the pieces kept whole?
When the charred white remains come of the oven on a oven trolley, most of the body is white dust except the larger bones. So to answer your question, they tend to keep pieces of delicate white bones in the urn, the dust is got rid of. There is some stuff on here on it. https://www.tofugu.com/japan/japanese-funeral/
Apologies if the questions sound naive, but it’s not something I’ve come across before.
Not naive, its a differet culture and who knows how a body burns?