What is the best way to allow users select items from a very long list in a responsive web app?

Scenario

I'm designing a web app where the user adds predefined items from List A to a custom List B.
List A is over 100 items sorted alphabetically.
List B can be customized according to user needs.

I designed a drag-and-drop solution for the web app. The user can jump to List A items alphabetically by clicking a letter.

drag and drop feature

Problem

This drag-and-drop feature doesn't work on tablets very well and is impossible for smartphones. So I designed a select form for mobile devices: When the user taps on a list item in List B, a picker/flipper populated with List A items is triggered.

mobile view select

Question

Is there a way to implement alphabetical filtering in a web app's mobile interface?

How can the mobile interface for this feature solve the following requirements?

  1. Allow user filter List A
  2. Add items from List A to List B
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