"To keep the conversation flowing" means to maintain a smooth and continuous exchange of ideas or information during a discussion, without any awkward pauses or interruptions. Full definition
Reading through past messages and picking out some things from their profile you'd like to know more about will help keep conversation flowing and allow you to be a happy confident date. (uk.match.com)
During the interview, keep the conversation flowing by inserting some of the thought - provoking questions you prepared beforehand. (aneliteresume.com)
Open - ended questions provide the best opportunity for people to elaborate on a given topic and will keep the conversation flowing smoothly. (entrepreneur.com)