The QA team lead will often remain but their role should change considerably.

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.

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.