B737 flight crew reported receiving airspeed disagree flag during cruise. Flight crew continued to receive unreliable airspeed indications as they continued to destination and landed.

Narrative

While at cruise flight from ZZZ to ZZZ1; we got a IAS Disagree flag on both Primary Flight Displays (PFD's). This went away within about 60 seconds but happened again about 4 more times. During these events; we noticed an IAS difference of about 5kts between the Captain and First Officer (FO) sides. I contacted dispatch and got a phone patch with Maintenance. I briefed Maintenance on the situation and they suggested we continue but monitor the instruments. During this time; we reviewed the airspeed unreliable checklist but we still were not sure who had the unreliable airspeed. After the third event; I opened the airspeed window and slowed down and then sped up about 15 knots each. We observed that the FO IAS indicator did not move but the Captain and standby indicators followed the speed changes appropriately; telling us that the FO side was unreliable. I then swapped rolls to become flying pilot and gave the radios and monitoring pilot duties to the FO. We ran the checklist for unreliable airspeed. Once we started our descent into ZZZ1; the FO airspeed indicator dropped to 45 kts and the FO altimeter stayed about 1;000 ft. higher than the Captains and the standby. Once we completely lost those indications; I made the decision to [request priority] and continue to land at ZZZ1. We informed the Flight Attendants and Dispatch of the emergency status and landed without incident.