Presidents are highly dependent on the team of experts with which they choose to surround themselves. Those experts have certain ideological beliefs. What is to be learned when the staff of that expert team changes in a time of crisis and new members challenge the earlier team's assumptions?
When presidents decide to overhaul their staff, including key policy advisors, they generally are looking to move their administrations in a slightly different direction. Most such staff purges -- assuming the personnel change is involuntary and the affected staff are not departing because they want a change; those types of jobs are extremely stressful and hard on families -- are the result of policy failures that diminish the president's public support. Those failures may involve the economy, the conduct of a war, or a major foreign policy failure. Because certain key personnel would be identified with the failed policy, the president would try to replace them with new people who are both philosophically compatible with the policy shift and have a record of loyalty to the president, the vice president, or some other high-ranking official within the administration.
It is rare for presidents to consciously appoint to important positions individuals with whom they are not in agreement on the issues for which they would be responsible. There are exceptions. President Jimmy Carter had key national security officials -- Secretary of State Cyrus Vance, Secretary of Defense Harold Brown, and National Security Advisor Zbigniew Brzezinski -- who were not only philosophically far apart, but temperamentally polar opposites as well, especially Brzezinski and Vance. The result was a dysfunctional national security team that was not able to serve President Carter particularly well, especially given Carter's own dysfunctional management style.
When presidents replace key staff, it is often because they want an ideological shift. They want different perspectives than those to which they had previously been exposed. Presidents are, without exception, strong-willed individuals. They want people who can get the job done while remaining studiously loyal. In the White House, loyalty often goes much farther than ideology, anyway. The president sets the agenda and the tone, the staff carries it out.