When you type something in the search the list of markers refreshes live and it seems like the markers refresh on the map itself but the map doesn’t re-center on where you are searching until you select a marker from the list.
Searching is ajax based when you search your any created location your respected location is automatically showed on your map.
Please check this tutorial https://wpgmp.flippercode.com/example/change-center-point-on-marker-click-in-google-maps/
I think it will helps you.
I gave you link of an example, please go to 'info-window settings' where you will find 'center the point', enable it then when you click on any marker at front-end your marker will shift to the center of the maps. Enclosed screenshots.