Dispatcher reported a flight crew did not respond to multiple messages providing the crew with alternate and divert airports due to deteriorating weather conditions in the arrival area.

Narrative

On Date midnight shift I had a flight [to] ZZZ on Aircraft X. ZZZ1 was listed as a destination alternate however at time of dispatch ZZZ was not requiring an alternate but there were some rain showers moving out. As the flight was halfway over the [body of water]; I received a WSI Fusion alert of ZZZ1 ceiling and visibility dropping. There was a tempo from 10/11Z of 1SM BR VV002. I looked at the TAF for ZZZ2 and saw that it was above mins. At the time; ZZZ2 was P6SM VCSH SCT003 SCT120 in the TAF until 14Z. The METAR was 1SM RXXL/P6000FT BR BKN003 BKN110 Date1/Date1 A2992. I sent an ACARS to the Captain alerting him of the falling ZZZ1 weather and that his alternate was changed to ZZZ2. I sent him fuel numbers; ZZZ2 weather and NOTAMS around XA:42Z. I received no response to the ACARS. As the flight got closer; ZZZ1 and ZZZ2 ceiling and visibility started dropping and fogging in. ZZZ was fogging in as well. As the flight was approaching ZZZ2; flights began holding in ZZZ due to RVR being around 5000. I sent an ACARS to the Captain at XB:35Z; alerting him to the holding and suggesting that we divert to ZZZ3 for a gas and go due to flight being tight on fuel. Once again; I received no response to this ACARS. The flight continued flying towards ZZZ. As the flight was lined up for landing on RWY XYL; the RVR dropped to 2000 prior to the final approach fix and the flight had to go missed. Since ZZZ1 and ZZZ2 were now fogged in as well; the Captain had to divert to ZZZ4. ZZZ4 had a tempo of BKN004; but since the flight was now tight on fuel it was the only viable airport left to divert to. Dispatch sent an ACARS to the flight at XC:06Z; instructing him that the flight needed to divert to ZZZ4 immediately. Once again; Dispatch received no response from the flight. The flight did divert to ZZZ4 and landed at XC:33Z. I had flights going through areas of moderate turbulence in the Midwest; so my attention was partially focused on them. I had nine (9) red-eyes in the air; so the workload and task saturation of flight following was high. When ZZZ1 weather dropped; the only airport I could fit on was ZZZ2 due to a heavy payload restricting the amount of fuel onboard. This in turn created more workload and task saturation. The lack of communication from the flight crew regarding my ACARS messages also contributed to my stress and workload issues.The Captain not responding to and possibly ignoring all ACARS messages from Dispatch about the deteriorating weather in south State is unacceptable. The Captain did not acknowledge the deteriorating ZZZ1 weather; ZZZ weather or ZZZ2 weather and new fuel burn and alternate change to ZZZ2. The Captain did not acknowledge Dispatch suggestion of diverting to ZZZ3 due to airborne holding/diversions in progress in ZZZ. The Captain did not acknowledge Dispatch ACARS of needing to divert to ZZZ4; when landing in ZZZ was not possible. Dispatch was kept in the dark of the thinking of the flight crew; Dispatch shares joint responsibility but in this case it seemed the Captain determined Dispatch was not on a need to know basis. This absolutely needs to change; maybe all ACARS messages need to be replied to/awk in the future.