Impeachment is analogous to indictment in regular court proceedings; trial by the other house is analogous to the trial before judge and jury in regular courts. Typically, the lower house of the legislature impeaches the official and the upper house conducts the trial.
At the federal level of government, agencies such as the U.S. Department of Justice's Civil Rights Division, the U.S. Department of Education's Office for Civil Rights, and the Equal Employment Opportunity Commission (EEOC) investigate claims of civil rights violations and discrimination.
For a more historical and international perspective, see Military aid to the civil power. Defense Support of Civil Authorities (DSCA) is the process by which United States military assets and personnel can be used to assist in missions normally carried out by civil authorities.