The fundamental way that an FPGA works (more or less) is to program the configuration registers with values that then determine the routing. One thing you could do is have multiple sets of these configuration registers, at process start you allocate one configuration to that process and set it up, from that point onwards you when you despatch instructions to the functional unit you do it along with the process configuration. So one process could be using the FPGA as a custom hashing function and another could be using it for some funky video IP. I'm not saying it's a guaranteed win but I think there are technical solutions that you could experiment with.