Key Lime pie filling thickens once the lime juice is added to the sweetened condensed milk and allowed to sit. Why does the filling thin out when it is pumped and require static hold time to thicken back up?
I've been making KLP commercially for well over 20-years and long before that growing up in Miami. Gone for me (on a commercial ever) are the days of using fresh egg yolks, but on occasion we might make one when visiting friends, from scratch. It is very important when incorporating the lime juice into the SCM-yolk mixture that it be done gently, never use a blender of mechanical mixer. You will see the chemical interaction between the yolks and juice start to take effect, "clumping" (for lack of a better word right now) will occur as the mixture begins to thicken. Also, the colder the ingredients, the better. If the mix is overworked, you go beyond that magic moment and it is very difficult to have the filling "set."
Another aspect is the juice. I would never recommend using a bottled juice over fresh, even if you're using fresh regular (Persian) limes. The most popular bottled juice used isn't even real key lime juice, but truthfully packaged as "Key West Lime Juice," concentrated lime juice packaged in Key West (not even any more.) The from-concentrate and reconstituted juice just doesn't hold up as well as (any) fresh juice.
On a commercial level, we now add (beef) gelatin to our mix, about 1/4 tablespoon per 10-inch pie. This is for insurance, as most of our customers are restaurants who need to cut and plate and cannot afford the sloppy-factor. The gelatin is blossomed in the juice then (under warm heat) after mixing with the juice it is allowed to cool. At that point the juice is incorporated into the SCM-yolk mix. This trace amount seems to be just enough to have the filling set right without adding any gumminess.
This answer is of course for the "fill and chill" method for making a KLP, as baking the pie once the filling is added causes the egg to cook and therefore sets the pie filling. Good luck! I hope this helped. Happy New Year!