Hello Kris, I would use one of next three ways to keep the inserts flat on foam base:
1. Glueing the pieces down onto foam base.
2. Most common way by using, 'listing wires'.
3. By using 'velcro material'.

Thank you. How would you insert the wires? From the outside stabbing them threw the foam?

The 'listing wires' would be inserted in the 'sleeves/bags' which go attached/sewn to the seams of vinyl/leather pieces.
These videos would give us an idea: https://www.youtube.com/watch?v=lEozir4Ngp0&t=55s
A basic seat cover (min. ten:06)https://www.youtube.com/watch?v=UR0Sx1DhZJA&t=2s
I hope it helps!

Done with Velcro. A technique I learned when doing corporate aircraft seating.

What I started with...

did you use the velco backed sew foam?

That top seat the 2 tone brown seat looks like a Snowden seat. I tried to contact them on how they keep it down but got no reply. Which is weird cause I may be wrong, but I feel they sell the seat frame and foam along with finished seats. Think they’d help out a potential customer.

I think he only sells supplies and uses a local upholstery shop to do his seats.