Air Carrier Captain reported an electrical burning smell from the forward equipment exhaust fan that was resolved by turning the switch off; enabling them to continue to their destination.

Narrative

Prior to departing ZZZZ the inbound crew had wrote up a status message for FWD EQ EXH FAN. This was checked by tech ops but no maintenance message was present. 2:10 into flight observed and felt a vibration in front of my control seat behind the display screens. Quickly lead to a smell of electric burning smell but no smoke observed. I would describe the intensity of the smell as light to moderate. The FO and I quickly turned to the QRH to focus on overheat of exhaust fans due to the location of where we felt and smelled the abnormal condition. I landed on FWD EQPT OVHT in the QRH which pointed to placing the fan to Standby. I asked the FO if that made sense to him logically and elected to use my CA Authority to attempt to resolve the malfunction prior to it getting worse and eventually leading to a full overheat. He agreed with my reasoning so we both confirmed the (un guarded) switch and moved it to standby. This stopped the vibration and stopped the increase of the electric burning smell. The smell eventually subsided over several minutes. While doing this I called via Sat Com to Dispatch and Maintenance Control to ask for advise and advise of our actions. The Maintenance controller thought that my action isolated the issue and that we were ok to continue. I wasn't there yet to commit to that so we had the relief FO weigh in on the event and decision which he supported. We also had a Line Check Pilot Captain (LCP) in the back of the plane riding home from my line check from the previous evening. I had the FA's wake up the LCP and asked him to come to the flight deck and help review our actions. He supported my decisions and also thought it was safe to continue to ZZZ1. Based on the fact that by the time we made the official decision to continue in ETOPS we were not far from the critical point.We continued without issue until landing where the fan logic switched with weight on wheels and we received a FWD EQ EXH FAN status message with the same moderate electrical smell as we taxied into the gate.At the gate we were met by tech ops who asked me to debrief and fill out the fume/smoke checklist so they could fix the issue. I also called the Chief Pilot on the way home to advise of my actions and was told to fill out an report. I called the outbound CA who was operating plane XXXX to advise of the issues we had and ensure he felt ok taking the airplane. The fan was determined to be failing and was replaced and the outbound flight was delayed by 4.5 hours.