I don't think this would apply to countries with a Parliamentary system, but at least in the US, have there been cases of a Republican running successfully as a Democrat or vice versa, an Independent as either, etc? I'd guess that there would at have been some of that in the 60's as the Southern politicians switched from D to R.
Actually, yes. It's not unheard of for a politician who loses the primary election to run in the general election as an independent or a member of another party.
Three examples immediately come to mind for me here in Alaska, and all happened within the last 30 years. In the 1990 election for Alaska governor, Arliss Sturgulewski defeated Wally Hickel in the Republican primary for governor. Hickel had served as the state's governor 20 years before and was selected by President Nixon to serve as Secretary of the Interior.
Having lost the primary, Hickel made an almost literally last-minute decision to run on the Alaskan Independence Party ticket. The Alaskan Independence Party is formally a secessionist party whose platform states that the original Alaska referendum on statehood is invalid because U.S. soldiers and other non-Alaska residents were allowed to vote on the issue.
Hickel won the election with a plurality of votes, defeating Republican and Democratic challengers. Two years after taking office on the AIP ticket, Hickel formally returned to the Republican Party.
The 2014 Alaska governor's race was structured similarly. Bill Walker initially ran as a Republican but switched to an independent ticket after joining forces with the Democratic candidate, who became his lieutenant governor. The "Unity Ticket" won the election by a margin of less than 5,000 votes.
In 2008, sitting U.S. Senator Lisa Murkowski (R-Alaska) was defeated in the Republican primary by Joe Miller, a lawyer from Fairbanks. Murkowski's defeat was attributed to a lackluster primary campaign and the effects of the tea party movement, which brought out large numbers of libertarian-leaning Republican voters.
Having lost the primary, Murkowski did not appear on the general election ballot but still managed to defeat both Miller and her Democratic opponent with a successful write-in campaign that encouraged voters to cast their votes for her. There's some debate as to whether this counts as a true third-party candidacy, however, for although Murkowski did not receive support from the Republican Party of Alaska, she never gave up her party affiliation, nor did she lose any seniority in the U.S. Senate. This was the first successful write-in campaign for the U.S. Senate anywhere in the United States since Strom Thurmond (D-South Carolina) did it in 1954.