Here are some photos of the inside of my barel box. There is a piece of felt on the bottom of the box in between the two center supports. There is also a piece of felt at the breech end which just goes to the top of the support. There does not appear to have been a felt on the muzzle end, nor any on the top.
The felt is white in the middle, but seems to have been impregnated on the outsides to make them grey. This box still had cosmolone/tar preservative on it, so that may be what coloured the felt.
I'll post some photos of the inside of the grenade boxes once I find it. I have one box which has the felt on the lid, as well as the wooden cradle for the 2 36M grenades.