Faced with bad weather and reduced RNP capability; a Dispatcher determined it best to advise a flight to divert to a station with better weather.

Narrative

Weather at planned destination [was] operational and above minimums for approach at departure time. Flight departed and weather at first alternate began to deteriorate prior to arrival [at] planned destination. [I] did not delete the first alternate as an alternate due to [it] still having legal alternate minimums. During this timeframe; also had shift/desk turnover situation explained to relieving Dispatcher turnover completed. A few minutes after turnover; aircraft had onboard equip malfunction and unable to continue the RNP approach to the planned destination. Captain called and said his fuel on board was 8.2; he was going to try another approach into the planned destination. I quickly scanned the new first alternate; the planned destination; and a second alternate's weather; and looked at the cameras that face the arrival into the planned destination. I determined that a missed approach would be a low fuel situation. The weather had deteriorated at the first alternate since his departure. It is less than 15 min; subsequently I asked the Captain to divert to the second alternate. The burn to the second alternate is 4.0.