nope, some moved out and some stayed. St Augistine Florida, founded 1565 and continuously inhabited.
You’re thinking of the South East where the Spanish came up through mexico and then left. Although desoto did go through the Tampa area and up into Alabama prior to that time.
Actually I’m hot on the trail of two Spanish settlements in the Ohio valley that may well date from DeSoto’s 1541 trip ~ but when a settlement is disrupted and vacant for a year or so, that’s an interruption in ‘continuous’ ~