Once you have registered on the padrón of your new home town, you can communicate your change of address to a list of authorities, through the electronic office of the Spanish government, all in one go.
Included are DGT/Trafico, TGSS, Seguridad Social INSS, Agencia Tributuaria (Tax Office) and the Dirección General de Policia, where the register with NIE numbers is held. You need a digital certificate, a Cl@ve Pin or Cl@ve Permanente to access the service. I tried with my digital certificate and it took a couple of tries to get through, so don´t give up if you get an error message the first time round.
The only disadvantage the system has, is a 2 month process time from your change of data on the padrón, so if you try to access the service before, your data might not have been updated yet.
On the other hand, I cannot think of many things being that time sensitive that this would be a problem and it is a whole lot more convenient than having to go in person, or visiting the various electronic offices one by one.
To communicate your change of address, choose the option on the left. To follow the progress of your communication, choose the option on the right.
You can even communicate your new address in Spain with the authorities in your hometown abroad, if you recently made the move to Spain. Not all European countries connected yet, but the list is growing steadily.