If you need to leave the apartment in order to register in another, you can simply file a passport for a residence permit at the passport office at your new place of residence. In this case, FMS employees independently sends a request to extract you from the apartment where you were prescribed before – regardless of what city it is. The only negative – the procedure statement and registration in this case will take quite a long time (two months).
If you want to check out "nowhere", not making permanent residence permit, to do it "in absentia" is much more difficult. Because to write you out of the apartment only on your personal statement, you will need to submit an application not removing you from the register and to assure him of a notary, and to issue the notarial power of attorney on the name of the person who will do the procedure of discharge from the flats for you. In addition to these documents, you will need to send the Trustee and your passport as it will bear the mark about removal from the register.
Despite the fact that by law, an extract from the apartment can be made by proxy, some passport offices (especially in big cities) refuse to do it. Therefore, before you issue a notarized statement and power of attorney, it is best to call the FMS at the place of your residence and to clarify whether they can write you "remotely".
If you live abroad, then to extract from the apartment in Russia to contact the Russian Consulate and make out a statement of deregistration. For registration statements are required to present a Russian passport.