DA40 Flight Instructor with student reported engine roughness and low fuel indications during cruise. Flight crew landed safely.

Narrative

This was a training mission. My student and I noticed a little bit of roughness when we started the aircraft originally. We thought it was most likely fouled spark plugs and we called our on call assistant flight training manager. We ran some tests and after a few minutes the roughness did not occur. Run up was normal and we decided to continue our mission (IFR flight in VFR conditions from ZZZ-ZZZ1-ZZZ). During the enroute phase of flight we noticed that a slight reduction to our mixture setting caused engine roughness sooner than usual; at this point we were closer to ZZZ1 so we decided to continue towards that airport and continued to monitor the situation. We were currently headed towards the IAF for the RNAV XX approach about 5 miles to go until we arrived at the fix when we got an indication for low fuel pressure. In addition to the low fuel pressure we then received a low fuel flow indication and then our EGTs rapidly dropped what looked like around 200-300 degrees. To me having multiple signs of fuel/engine problems indicated to me that we might have a possible full or partial engine failure. We were on an instrument flight plan however the conditions were vfr. Because of the warnings and the environmental allowance we decided the safest course of action would be to head directly towards the airport rather than continue flying outbound towards the fix. We let ZZZ center know we were potentially having engine problems and that we needed to turn directly towards the airport and land immediately. They cleared us for the visual approach and we continued inbound. ZZZ center asked if we wanted to request priority handling and initially we declined; however; we changed our minds and did just to be on the safe side. We were able to land safely at ZZZ1 and contacted Center on the ground to let them know we were alright and clear of the runway parked at the FBO.