I was doing tests with the <input type="search"…> tag in HTML5 on my website and it works perfectly. There is only one small detail and I only saw it in the browser Mozilla Firefox (not in Chrome , IE / Edge or Opera )
It happens that when viewing this text field and writing the phrase to be searched in the field, in Firefox the "X" does not appear on the right to delete the content that was put in the text box. I thought that the problem was only mine, but it was given to me to prove what happened with the search engines of the search engines ( Google and Bing ) and it was the same said > "X" does not appear either.
You can try it yourself (if you have Firefox and compare it with another one). Someone knows the reason for this little detail in the Mozilla browser. I do not think any attribute is missing or you have to add a particular style.