This post seems to be related: Using the JavaScript embed with ASP.net Forms
The post shows that the default behavior of asp.NET is prevented out of the box by Swiftype.
However, the question still remains. Since the documentation of Swiftype shows that you need to add a form element around the input and button in order to make it work, as you can read here: Customizing Swiftype Visual Styles | Swiftype Documentation
But since Sharepoint adds one big form around the entire document, then that limits your option if you’d for instance there is a action assigned tot the asp form (always by default) want two Swiftype engines on the same page.
The article states that:
ADDING A SUBMIT BUTTON TO YOUR SEARCH BOX
Who doesn’t like buttons? To add a submit button to your search input, you can wrap the input code in a form container element and add a button element. All styling of the submit button is up to you! You can find an example below.
<form>
<input type="text" class="st-default-search-input">
<button type="submit">Submit</button>
</form>