Furry Technologists

There are a few ways to do it for Xilinx/AMD For Zynqs, the processor actually programs the programmable logic, so you just need bit files in the OS file system and your good to go. For any part there is also partial reconfiguration where small bits can be programmed with alternate partial bitstreams without reconfiguring the whole device. There are a bunch of conditions that have to be met, and I don't have any experience with that style of design, but yep, self reconfiguration, at leas on Xilinx parts, is definitely a thing.