Run this script on https://www.linkedin.com/messaging/compose/ to empty your messages inbox. You might need to run it more than once.

delete-all-linkedin-messages.js

(($) => {
  'use strict';

  const click = button => new Promise(resolve => {
    $(button).click(resolve).trigger('click');
  });

  const sleep = timeout => new Promise(resolve => setTimeout(resolve, timeout));

  const process = async () => {
    const conversationListItems = $('.msg-conversation-listitem__link.msg-conversations-container__convo-item-link');
    if (conversationListItems.length) {
      await conversationListItems.toArray().reduce(
        (promise, conversationListItem) => promise.then(async () => {
          await click(conversationListItem);
          const dotButton = $('.msg-thread-actions__dropdown.ember-view')[0];
          await click(dotButton);
          const deleteButton = $('.msg-thread-actions__dropdown-options.artdeco-dropdown-with-arrow.ember-view ul artdeco-dropdown-item')[1];
          await click(deleteButton);
          const yesDeleteButton = $('.artdeco-button.artdeco-button--2.artdeco-button--primary.ember-view')[0];
          await click(yesDeleteButton);
          await sleep(500);
        }),
        Promise.resolve(),
      );
      await sleep(2000);
      await process();
    }
  };

  process();
})(jQuery);