There are ample documents on how to structure a brainstorming session. Google brainstorming. Recent research has shown that brainstorming doesn't actually work very well, but your boss has assigned you to do this, so facts don't really matter. The key rule of brainstorming is that there must be no criticism, and that all contributions are taken seriously.
There are two general strategies to enforce the brainstorming rule. The first strategy is for you as the moderator to step on any criticism fast and potentially hard. Make it clear that contributions are welcome and criticisms are not welcome. Friendly, respectful edits & clarification are ok. The second strategy is to democratize the responsibility. Our team keeps a set of yellow soccer flags in a bowl on the table; if anyone feels that anyone else is being disrespectful/critical, anyone can throw a flag on the play. The second strategy requires a more mature, more trusting team. The first strategy requires you to bring your A game and may require you to stop recording and focus on moderating.
There is too little information to answer the question. In a comment (which you really should edit back into the question, so that we have all the information necessary) you mention that there will be six people from two services. I would probably structure the meeting this way:
10 minutes establishing the goal of the process - ensure that everyone has a shared vision of the outcome including any relevant quality standards.
15-20 minutes documenting the current process. I'd probably force each team to document the other team's process with an emphasis on the interfaces between the teams and between the teams and the external stakeholders.
10-15 minutes documenting opportunities for improvement - identify the pain points. Impose a rigid "no response" rule - nobody gets to defend the status quo, nobody gets to criticize or offer feedback. The goal is to capture places where at least one person believes that the current process is painful. If one person believes it is painful and five people believe it is optimal, then the group must accept it as painful.
15-20 minutes on optimization - Once again, standard brainstorming rules, no criticism, no defense. Crazy ideas permitted, unworkable ideas encouraged.
Select and optimize the best candidates for the remaining of the meeting. Which of the options has the best chance of success? Which is the quickest win? Which has the greatest long term potential to optimize? Which can only succeed with strong executive sponsorship?
10 min Debrief and establish an agenda for future work. Who is accountable for doing what, and how will the action be briefed?