The QA team lead will often remain but their role should change considerably.
They will now focus on supporting the individuals instead of a project
Instead of being focused on specific current project topics, issues and deadlines the lead should switch to a role that supports the employee more generally in terms of skills development and career advancement, as well as engaging senior management in support of these goals.
What you will hear a lot for this change (as already indicated in your question title) is that management is now "cross-functional". This means that for project specific issues you are effectively 'reporting to' to the project owner or lead (actually to the team is a better approach). For employee specific issues such as career training, promotion, etc. you use a QA lead.
The team lead is likely to find this quite hard. They are essentially going from a 'supervisory' to a 'support and encourage' role and this requires really different skill sets and approach. Many companies miss this and end up not actually changing their organization, just using names such as 'stand-up', 'retro', 'backlog grooming' to persuade themselves that they have changed but outside of those ceremonies the practices throughout the day may not have changed. Change is really hard and change from supervisor to councilor is one of the hardest.
They will also focus on new aspects such as test training, cross-training, etc. which will allow the lead to use folks more interchangeably in more of an 'internal consulting model' allowing for both more flexibility to meet team needs as well as more opportunity to learn and grow for individuals who want it.
Finally, the team lead can focus more on the organizational goals for quality and spend more time engaging senior management to get the support and backing for quality goals and initiatives.