This is the approach I'd take.
Remove all dust from the joining pieces and surfaces.
Make a dry-fit, before glue-up, aligning and mating the wood pieces and insure the miter joints are all closed.
Examine all corners and sides, front a back.
Use a strip of masking tape to hold the wood pieces to the stone, while doing this exercise.
Clean the surfaces to be mated and glued, using rubbing alcohol.
The rubbing will clean the residue from the stone and wood, permitting for more secure bonding.
I'd use a Clear Silicon Rubber Adhesive/Caulk.
Lay the bead in the rabbet joint on the wood, press the wood to the stone, and clamp in place.
The mitered corners, when assembled and fitted to the stone, can be secured using a pinner (and air driven tool that uses a very thin headless brad nail).
Clamping can be accomplished by banding clamps, located near each corner, in both directions, and a band across the center in both directions.
During clamping, insure all joints are closed, and the wood frame is resting tightly against the stone, with no visible cracks.
The Silicon rubber will allow longer working time, while positioning and clamping.
Wipe-up/clean-up of any visible silicon adhesive can be done after the adhesive has set, by using a scraper, such as a single-edge razon blade fitted in an appropriate holder.
NOTE: apply an applicable finish on the wood, onoly to the surfaces that will not be exposed to the adhesive.
The adhesive will not stick to an oily substance on the wood.
Thanks for the response and great ideas. What clamps do you use for something like this? I'm a bit nervous about using brad nail as to chance of splitting wood.
Thanks for your reply.
#1 band and web clamps, may also be known as "adjustable strap clamp".
A webbing material that features a clamp to draw-up and tighten as well as loosen the pressure for release.
Look at the "band clamps for woodworking" on Amazon, to give you some ideas what may work for you.
#2 rubber-band material. I found some large size rubber bands at Rockler, for similar tasks, that worked very well.
Bike inner tubes work very well too. The material is stretchy---can be adjusted to the desired pressure needed for a specific clamping task.
Used bike inner tubes can be obtained from bike shops.
Auto/truck tire inner tubes can be cut into strips, to make large rubber bands.
TIP: When cutting into strips, make accurate width cuts, so the cross-section is even. If an area is too narrow, it'll tear out before the wider section fails.
#3 large bar clamps. When using these type tools, insure that your workpiece surfaces are protected/padded, so as to avoid marring/damage to the wood. Padding, such as cardboard will suffice.
When clamping, apply just enough pressure to close the joint.
Excessive pressure will damage your work-piece and may render it unrepairable or useless.
A "pinner", is a "pin nailer"---a 23ga headless nail, as thin as a "pin", and driven by an air-nailer.
One thing one has to be carefull with, is that when the pin is driven-in, at the rapid rate by the air-nailer tool, it will followed along the grain-line in the wood and may blow-thru the back side, which may cause damage to your finished work-piece.
Beginners may not be comfortable with this approach, so put this approach aside for now.
I love pinners for such tasks, but, it does require skill, to fasten small cross-section items.
I've also used "quick-grab" adhesive. It sets up fast.
Place a small amount on the mitered joint ends, and press and hold the joint together for the time denoted on the instructions. You can also apply some masking tape over the joined area, while the adhesive sets, so you don't have hold it by hand.
Hope this helps.