how to add a redirection after password change without modifying Drupal Core?
You can add your own submit handler to the form using hook_form_alter, and set your redirect here.

Ex:

&lt;?php
function my_module_form_alter(&amp;$form, &amp;$form_state, $form_id)
{
  if($form_id == 'user_profile_form')
  {
    $form['#submit'][] = 'my_module_profile_form_submit';
  }
}

function my_module_profile_form_submit($form, &amp;$form_state)
{
  $form_state['redirect'] = 'some/path';
}
?&gt;