Citizens living abroad can register in person or through electronic registration. They receive at home an envelope with the ballot and the voters will send the ballot to the embassy or consulate (Source). See the sources for the quotes from the law providing for external and postal voting.
Hungary, Act XXXVI of 2013 on election procedure [as in force on 20 November 2021], accessed 10 April 2022
Section 267/A. The National Election Office shall make available the part of postal voter electoral register that contains those voters who requested the receipt in person of the postal voting package by the sixteenth day before voting day, and it shall transfer it, applying the provisions of section 263, to the competent election office.
Section 268 (1) A voter recorded in the postal voter electoral register may request, in person, information at the diplomatic mission about data recorded under point p) pa) and pb) of Annex 2 and point c) cd) of Annex 8. After verifying the identity of the voter, the diplomatic mission shall record the request in minutes and attach it to the data request sent to the National Election Office.
(2) The diplomatic mission shall obtain the data referred to in paragraph (1) directly via the National Election Office, and inform the voter thereof.
120. Voting at a diplomatic mission
Section 269 (1) Voting at diplomatic missions shall be possible if the host state does not object to it.
(2) Votes may be cast at diplomatic missions on the voting day in Hungary between 6.00 and 19.00 local time. At diplomatic missions where the time difference is -1 or -2 hours compared to Central European Time, votes may be cast between 6.00 local time and 19.00 CET. At diplomatic missions established on the American continent, votes may be cast on the day before the voting day in Hungary between 6.00 and 19.00 local time.
(3) With the exception of vote counting, the diplomatic mission election office shall carry out the tasks set out for the polling station commission. The head of the diplomatic mission election office shall decide any disputed matters arising during the voting.
(4) Voters recorded in the diplomatic mission electoral register may vote at the diplomatic mission or, if the building of the diplomatic mission is not suitable for conducting the voting, at another premises provided by the diplomatic mission. If necessary, more than one premises shall be provided for voting, and a sufficient number of polling booths and ballot boxes shall be provided to ensure continuous voting for the voters recorded in the diplomatic mission electoral register.
(5) From the start of voting to the closing of voting, only members of the diplomatic mission election office, media content provider representatives, observers at diplomatic missions and international observers may be present in the polling station.
(6) The National Election Office shall be informed of the suspension of voting without delay with a view to ensuring the lawful continuation of voting.
(7) The diplomatic mission election office shall place a control sheet in the ballot box containing the time and date of the placement of the control sheet and the signatures of the members of the diplomatic mission election office present and of the voter voting first.
See further Sections 270-280 for more details of OCV regulations.