I understand at a high level how the Mixed Member system used in German general elections works during the general election.
Pages such as Electoral Reform cover this area reasonably well.
What I don't know and haven't been able to find is how MPs are replaced if they die, resign or are otherwise removed from office.
Yes, Germany does hold by-elections in the case when a constituency nominee dies between being nominated and the election, but when an MP dies or withdraws from the Bundestag they are simply replaced by a member of their party from the Land list at the last election - or the seat remains vacant. These scenarios are all governed by the Bundeswahlgesetz (Federal Election Act), an older version of which is available in English here.
Section 48 of the Act provides the general course of action:
If an elected candidate dies or informs the Land Returning Officer in writing that he refuses to accept membership or if a member dies or later withdraws from the German Bundestag for any other reason, the vacant seat shall be filled by a candidate from the Land list of that party for which the elected candidate or former member stood at the election. This shall not apply as long as the party has any seats as per Section 6 subsection (6), fourth sentence, in the respective Land. When a successor is to be selected, any candidates on the list who have resigned from the party after the Land list was drawn up or have become members of another party shall not be taken into consideration. Candidates on the list who as elected constituency candidates have refused membership or have resigned as members of the German Bundestag shall also be disregarded. If the list is exhausted, the seat shall remain vacant. The decision as to which candidate from the list is to succeed to the seat shall be taken by the Land Returning Officer. He shall notify the successor from the list and invite him or her to state in writing within a week whether he or she accepts election.
This has taken place four times so far in the 20th Bundestag:- Michael Sacher replaced the resigning Oliver Krischer, Daniel Rinkert replaced Rainer Keller after the latter's death, Nils Gründer replaced the resigning Thomas Sattelberger, and Clara Bünger replaced the resigning Katja Kipping.
It used to be that if the MP being replaced belonged to a party not able to submit a Land list, a replacement election was held within 60 days, unless there would have been a general election within the next six months - in which case the seat remained vacant. In 2023, this was changed so that the seat always remains vacant.
Separately, a by-election is held if the conditions laid out in Section 43 of the law are met; that is that a constituency candidate dies after the approval of the constituency nomination, but before the election.
A supplement to CDJB's answer:
The term "By-Election" for German "Nachwahl" is used in this older English version of the Bundeswahlgesetz. But there are some peculiarites that might make the translation misleading.
(The translation is older, but the text of Section 43 has not changed since then.)
Section 43 By-Election
(1) A by-election shall take place
- if an election has not been held in a constituency or a polling district,
- if a constituency candidate dies after the approval of the constituency nomination but before the election.
(2) The by-election shall take place not later than three weeks after the day of the general election if subsection (1) number 1 applies. It may be held on the day of the general election and shall take place not later than six weeks after the day of the general election if subsection (1) number 2 applies. The date of the by-election shall be set by the Land Returning Officer.
(3) The by-election shall take place in accordance with the same regulations and on the same basis as the general election.
(4) In the case of a by-election, the provisional result of the general election shall be determined, established and released immediately after the general election has been held on the basis of the votes cast.
Either there was no original election held (for whatever reasons), or the "By-Election" is held within six weeks of general election day. In this case (when a candidate dies before election day), the original election might have been taken place, but only the provisional result will be published. A consequence of this is that the Federal Returning Officer will never notify the winner of that first election, and they will not attain membership of the German Bundestag (§ 42 subsection (2) and (3)).
The By-Election therefore does not replace a member of the Bundestag, but fills an initially empty seat.
Applications of that rule are rare. Wikipedia cites three occurences since 1949, the last one in the Dresden I electoral constituency in 2005. That election caused a public discussion about the theoretical possibility of negative vote weight. (It did not occur, but because of the Apportionment paradox a member of the CDU in Northrhine-Westphalia lost his seat in favor of a member of the same party in Saarland.) Also, the provision in subsection (4) about the publication of provisional results was critizised as creating an inequality between voters, as the Dresden electorate had more knowledge about the outcome of the general election than any other voters.
