Based on the insights from executive coach Ed Batista, supporting a team member who is underperforming after coaching and will be demoted involves several key steps to ensure the process is handled with care and respect for the individual:

1. Data: Ensure there is clear evidence of a performance gap, and share this data with the underperforming team member. It is important that both you and the team member recognize that a gap exists and that, at present, they are unable to fulfill the requirements of their role. This should be an ongoing process so that when a change is necessary, it doesn't come as a surprise to the employee [The Fine Art of Levelling].

2. Non-Defensiveness: Encourage the team member to approach the situation without defensiveness. This quality can be developed by regularly stretching employees out of their comfort zones and fostering a culture where people are comfortable speaking up about issues [The Fine Art of Levelling].

3. Trust: The team member must trust your intentions, even if they disagree with your judgment. This trust is built over time through consistent efforts to show care and interest in employees as individuals. Be mindful of how you create safety or inadvertently pose a threat to employees, and work to minimize the risk of a threat response [The Fine Art of Levelling].

4. Learning: The team member should see the opportunity to learn from their new manager or the new arrangement. This depends on the leader's ability to hire a manager who is not only experienced but also capable of imparting knowledge to others [The Fine Art of Levelling].

5. Role Clarity: Clearly define the team member's new role and the distinctions between that role and their former role. Manage expectations about their membership and participation in various groups within the organization, and prepare to have difficult conversations about these changes early on [The Fine Art of Levelling].

6. Grace: Handle every aspect of the demotion process gracefully, ensuring that the team member feels respected and appreciated. This includes being honest about their performance, offering fairness in the process, and preserving their dignity [Merciful Exits (On Under-Performing Executives)].

7. Feedback and Coaching: Provide direct and candid critical feedback regarding performance. Explore more effective coaching and feedback methods, and consider other potential interventions and means of gathering data [Merciful Exits (On Under-Performing Executives)].

8. Cultural Considerations: Be aware of the organizational culture and how it might have contributed to the situation. Avoid scapegoating the individual for broader issues and consider what the rest of the team can learn from this experience [On Firing a Senior Team Member].

9. Communication: Communicate the decision and its rationale clearly to the team member and prepare for their emotional response. Do not present a mass of evidence justifying the decision, and avoid leaving room for the person to advocate for their retention [On Firing a Senior Team Member].

10. Support Transition: After the demotion, continue to support the team member in transitioning to their new role. Ensure they have the necessary resources and guidance to succeed in their adjusted position.

Remember, the goal is to manage the demotion in a way that maintains the team member's self-respect and minimizes disruption to the organization while also addressing the performance issues at hand.