Let it switch automatically to the next iteration when it is
necessary.
I'm afraid this could not be achieve now. Because initially, the design logic of dashboard is let you and your teams share information, monitor the WITs progress and trends flexibility with one highly-configurable dashboards.
Ever has the same puzzle and demand with you, after discuss more with the engineer from our Boards team, I clearly got the reason for this design.
As normal, dashboards is used for yourself or team leader to monitor the develop process. If we simply set the time period updated with the sprint automatically to display the data, this is not very helpful for tracking the staged data. Because if the overall data looks very perfect, it will let the user easily ignore potential problems that exist within a certain period of time. Providing highly-configurable widgets in the dashboard, can very convenient for members to better track their development progress with the phased custom time period data analysis. Also, This is more conducive to the leader to find problems in time.
For this, you can refer to this doc: About dashboards.
Gain visibility into your team's progress by adding one or more
widgets or charts to your dashboard. Customizable, highly-configurable
dashboards provide you and your teams with the flexibility to share
information, monitor progress and trends, and improve your workflow
processes. Each team can tailor their dashboards to share information
and monitor their progress.
Though this demand could not be achieve in dashboards, but our PMs has considered the convenience of this demands in many cases and taken it into Azure Devops. In the Analytics tab of Sprints pannel, you can view the Burndown chart within one iteration.

Get insights into your team’s health with three new Azure Boards reports
There has one feature request raised on our official uservoice community: Burndown Widget: Allow @CurrentIteration to be used. If this feature could be expanded into widget, it can give you what you want.
To driving this new feature developed, you can vote and comment it there. The corresponding Product Team will review these tickets regularly, then task a consider about expand it into our roadmap.