When searching for a template in the Transclusion dialog, a list of possible matches is displayed which can be navigated using your keyboard. Pressing the Tab key on one of these should 'Select' the template for use rather than close the inspector.
Tested at: https://en.wikipedia.org/wiki/Special:PagesWithProp
Alternative native select: https://developer.mozilla.org/en-US/docs/Web/HTML/Element/select (CodePen doesn't support IE 11, another alternative: https://www.w3schools.com/tags/tryit.asp?filename=tryhtml_select)
Procedure: tab into the select box, press space once to open the dropdown , press down arrow once, and then:
|pressed key||Opera 12||Blink/Win||Chrome 83/Win||Chrome 83/macOS||Firefox/Win+macOS||IE 11||Saf||OOUI v0.39.0|
Legend: = confirms choice, = cancels choice, — = does nothing
Text input suggestions
Tested at: http://davidwalsh.name/demo/datalist.php
Procedure: tab into the search box, type 'Mo', press down arrow once, and then:
|pressed key||Opera 12||Blink||Firefox||IE 11||OOUI|
Legend: = confirms choice, = cancels choice
 Opera 12 immediately places the suggestion into the text input. One can press the up arrow to cancel, in a way.
 Otherwise one can use arrow keys to change the value of the closed dropdown.
 Firefox doesn't let me open the dropdown with keyboard at all… I tested after clicking it with a mouse to open.
 Browser immediately chooses the highlighted option, leaving no way to cancel the choice.
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