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You can use hook_form_alter or hook_form_FORM_ID_alter to alter a form.
Core's search block form has a form id of search_block_form, so you could implement with hook_form_FORM_ID_alter() like:
use Drupal\Core\Form\FormStateInterface;
function mymodule_form_search_block_form_alter(&$form, FormStateInterface $form_state) {
$form['#action'] = '/my-custom-...
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