Bachuss's explanation is essentially what I learned many years ago .
If the joint is totally filled, a three sided fill, the caulk will tear away over time from one or both sides.
THe idea is to get a two sided, opposite sides, join. On a 90o joint it is needed to put something in the very corner as a bead breaker.
You may also need to lightly sand the are you want the caulk to bond to. FIRST dewax, then sand lightly , then dewax again. A trick I have read about from boat hull painters. If you just sand first you may drive any wax into the surface and then when you solvent clean the roughness will hold some of the wax/dirt.
The final join is very dependent upon how clean you got the surfaces. Any wax, any dirt, will interfere with a good bond.
I have not had to fill a join like yours is described but I think I would first try the foam backing rods available at the local building supply. THey can be purchased in several different diameters.
I'll add that I use butyl tape but have not for anything like described here. However for working it I use scissors to cut strips where needed, narrowing the pices and shortening as needed. It will bond to itself thoroughly when the final press is done.
Several strips could be worked into place without too much fighting where one bigger one may be a fight. Use a tool, several, made of polyethylene. The butyl will stick less tenaciously to that than to other materials. If the butyl can be kept cool it will be easier to work into place. Ice box to cool it. Once it warms it can get a final pressing into place.
Cleanliness of the area will also be important even with the butyl.
For that final pressing use some of what Commodave suggested, the backing tape., Maybe waxed paper .