The verb unseat (“To deprive of the right to sit in a legislative body, as for fraud in election, or simply by defeating them in an election” — en.wiktionary) is sometimes used in this context.
Following are some quotes from entries in the 1996-2008 webpage link at the
ngrams for unseat webpage:
• Houston Democrat John Martinez is trying to unseat incumbent John Culberson (R). Educator Felix Alvarado (D) is challenging incumbent U.S. Rep. Kay Granger (R). Business owner Lico Reyes will face incumbent U.S. Rep. Michael Burgess.
• Speaker of the legislature and member of the People's Party of Armenia Armen Khachatrian resigned his position as speaker because of efforts to unseat him from the post.
• After a lull of nearly a year, the opposition Movement for Democratic Change (MDC) has resumed its bid to unseat President Robert Mugabe from power, this time not through the ballot box but via mass protests ...
• For an internship in a course on Christianity and politics, I immersed myself in a challenger's race to unseat a strong incumbent senator. The race was tight and high profile for the ever-strengthening Republican Party in the South...