I've made a few cabinet doors over the past year, but they were fairly simplistic. I dado'd out a slot in the back and glued a panel into it. It was for cellar cabinets and style/quality wasn't as critical.

For a recent project, I wanted to make a higher-grade door, so I bought a set of router bits to make a floating panel door. The bits work great, but the problem I have is that the floating panel must be slid into place while one of the frame pieces is removed.

With my cellar cabinet doors, I was able to completely assemble and glue the outer frame, and then sand the joined sections so that they were perfectly flat and nearly seamless prior to finishing. With the floating panel doors, I can't do that. I have to stain and poly them before assembly. Once assembled, I put a final coat of poly on the outer frame, but I see no way to make it a perfect joint this way.

Is there a "correct" way to do this? I can't see how you can stain and poly it after you assemble it, because the poly is definitely going to seep in between the frame and the "floating" panel, eliminating its ability to float.

Sand all the profiles and the back of the panels that will be difficult to impossible to get after assembly.
Sand the inner profiles on stiles and rails also.
Assemble the doors with glue, then sand the flat sides of stiles and rails, flushing up joints and flatten center front of panels.
Take a common lead pencil and draw zig-zag lines across joints, when sanded high spots disappear first, sand evenly until all pencil marks gone.
Finish and hang doors.

So you're saying you assemble it completely before finishing? How do you keep the poly from sticking the rails/stiles to the panel?

I have worked in probably 20 shops in 40 years, including 4 of my own.
This is the way everyone I know does it, (sane people anyway).
Mennonites here in Belize assemble wood doors without sanding, it is near impossible to get them sanded after. Many problems, cross-grain sanding on tops of raised panel profiles is just one of the major issues.
They also use Mahogany sapwood, as all the heartwood is exported.
When you hear about Honduran mahogany this is where it came from.
Belize was formerly British Honduras.
In short they make things for poor people, you get what you pay for.
To locals it serves it's purpose, not a thing of beauty I can assure you.
The sticking of finish has never been an issue that I know of.
The wood will move if ambient moisture changes and break any adhering with no ill effects that i have seen.
For the record I don't use poly unless outdoor kitchen, I like lacquer.

Stain and poly the panels before installing them in the frame. You didn't say what the panels are made of but if they are solid they may shrink/swell/move so prefinishing will ensure that any movement won't show unfinished panel edges. If you poly the frame after installing the panel the amount of poly that may run in from the frame will not be enough to prevent the panel from moving if it wants to.

Thanks. I made the frame out of red oak. I couldn't find any thin red oak panels, so I went with birch plywood. After staining, it looked identical to the red oak.

What I was thinking I might do next time (and this is probably what you're suggesting) is to completely finish the panels AND the inner edges of the frame. Assemble and glue it, then sand and finish the rest of the frame. That way, I won't have to touch the area near the panel.

Should have mentioned that I spray all my cabinetry & furniture.
If you want a presentable finish, this is a must.
The only way you get runs is overspray or trying to apply with a brush, never works for fine woodwork.
I also use wood that is fully dried, so there is only one way the wood will move.
The panels may get wider, never narrower, so the 1/4" in the slots with 1/8" for expansion will never show.
Finishing the panels before assembly will work fine.
But the care with which you will have to handle them when later assembling and sanding stiles and rails without nicking the panels with sander precluded this is any shop I've ever worked in.

Plywood panels don't need to float. They can be glued.

Plywood doesn't expand/contract.

When I make solid wood panels for cabinetry, I use Space Balls in the grooves that the panels fit into. Two on each edge. See the maple hutch project in my gallery here. That piece is over 15 years old now and the panels have not moved at all. I prefinish the panels prior to assembly so there will be no exposed raw wood if they do move later on. I dry fit the frames and sand to the next to final grit. After assembly, final sanding is minimal and can be done by hand to avoid cross grain scratching. I use wipe on varnish to finish my projects. I had a problem with a factory finished six raised panel door on the pantry of our new home. All six panels shrank after about 6 months, exposing raw wood on all of the vertical sides. Fortunately, the painting crew left me a gallon of stain from doing the windows and I was able to color the raw wood in with an artist's brush.

If you have the time there is some benefit of staining the panels before assembling. With wood movement occasionally a panel will shrink and you will see a line down the edge with unfinished wood. I wouldn't go so far as applying a finish. Sometimes the panels will get dirty or scratched with something and need a little sanding. If it had a finish on it, it would further complicate the finishing. Also with stain it's usually better to stain everything on the same day. The weather can affect the color. Anyway for myself I will sand the panels but assemble everything unfinished and then stain and finish it later. The finish won't glue in the panels enough to prevent movement.