If you have to conduct an interview over a meal, eat first, then when the food is gone and the plates have been taken away, conduct the interview. The progressive interview is the process where candidates are interviewed in rounds. The first round is generally conducted by someone whose responsibility it is to screen the initial set of candidates. When all candidates have been screened, a group is selected to return for second interviews, normally with the hiring manager. When the second interviews are completed, finalists are normally invited back to meet other members of the team, other managers whose opinions are valued, the department head, the president, what have you.
Progressive interviews offer the advantage of structure.
The interviews progress with the presumption that those candidates who survive all of the rounds have leapt all of the hurdles and can do the job. Candidates who stick with the process are clearly interested. The disadvantage of progressive interviews is that they can sometimes take weeks to complete, depending on the number of rounds and the availability of the interviewers and the candidates. If a week or so passes between interviews, it's often easy to forget who said what. The team interview is a process where a team of interviewers is assembled that consists of parties who have a close interest in the open position.