Here is a solution for creating an instant delayed search:
Javascript Instant Search Feature
Just change your JavaScript-Search-Snippet as follows:
$this->registerJs(
'function instantSearch() {
$.pjax.reload({container:"#countries"}); //Reload GridView
}
var timer;
$("document").ready(function(){
//$("#new_country").on("pjax:end", function() {
$("input").keyUp(function(){
timer && clearTimeout(timer);
timer = setTimeout(instantSearch, 200);
});
});'
);