Most newer Intel (Cyclone and beyond) and Xilinx (-7 series) support various levels of partial logic reconfiguration. If you plan your device layout right, you can simply reprogram a region of fabric that is hardwired wired to the 'shell' of your system (which is connected to the peripherials). If you have setups like Vivado's SDSoC/high-level synthesis, you can do things like dynamically reconfigure OpenCL kernels, etc.

Of course, the problem is synthesis, verification, testing etc is all rather difficult in practice and takes a long time. The boards themselves also can't do synthesis/compilation, obviously, since they're fairly limited and it's a time-consuming process. You'd have to make it act more like an OTA update system, but that comes with a lot of its own problems...