What was the range of white responses to African American activism at the local and national levels?

I'll focus my answer on the Civil Rights Era, which began in the mid-1950s and reached its peak in the late-1960s. Most advances were made at this time due to there being a great deal of political activism, particularly in the South.

On the national level, the response to Civil Rights activism varied depending on who was president. Eisenhower had unwittingly seated Earl Warren as Chief Justice of the Supreme Court. Warren was a moderate who, with his role in the Brown vs. Board of Education decision, would become a leading figure in the effort to integrate public institutions. His own efforts toward Civil Rights were modest. Like Truman, he promoted an effort to discourage segregation in government contract jobs, but did not make many inroads beyond that.

President Kennedy was mostly concerned with foreign policy. However, in 1963, his position shifted. He was horrified by the police's violent reactions to protesters in Birmingham, and by the violence with which Freedom Riders were met by Mississippians. He began work on a civil rights bill, which would be signed into an act by President Johnson in 1964. 

Johnson also signed the Voting Rights Act into law in 1965. While he produced the most legislation, civil rights leaders, including Martin Luther King, Jr., were wary of him. Johnson was a political animal: if civil rights were going to inconvenience him politically, they could wait. King, however, refused to wait and used his national and political influence to pressure Johnson into signing the Civil Rights Bill into law.

Richard Nixon had no discernible interest in civil rights issues. In fact, one could argue that he helped to worsen things for black people and other people of color when he instituted the War on Drugs in 1970. His "war," supported by legislators such as Senator Nelson Rockefeller in New York, sent small drug offenders, usually users, to prison for years at a time, making them unemployable when they returned to society.

At the local level, responses to black activism depended on the state. The South is best-known for its aversion to any efforts to make things equal for people of all races. However, states and cities in other parts of the country also played a role in continuing inequality. 

Efforts to integrate schools through busing programs were resisted in both Boston and Seattle. Chicago was notorious for red-lining. This was a form of discrimination in which black people were denied mortgages in parts of town that were inhabited by whites. Moreover, the mortgages that they acquired were often much higher than what a white person would pay for a house of a similar size in a similar neighborhood. These unfair mortgaging practices not only limited people's abilities to choose their communities, but also made it so that they spent nearly their entire lives paying off a loan on a house.

In sum, white responses to black activism on the national level wavered mostly between indifference and reluctance. On the local level, depending on the state, white responses could be violently hostile, or insidious and indirect.