How to run lubuntu minimal from usb with persistence?

A netboot (minimal) .iso lacks the live "Try Ubuntu" environment that you would change and save. It's simply a bare-bones installer.

If you wish to customize the netboot .iso, that's very different. It involves preseeding or unpacking/repacking the .iso, and has nothing to do with persistence.

The ubuntu documentation gives instructions on how to install lubuntu minimal.

This is completely unrelated to the Live USB persistence. The instructions on that page are for installing ubuntu from mini.iso. This mini.iso contains only installer and not Live (Try Ubuntu) option. Hence it is not possible to run minimal lubuntu.

Adding Persistence to a Lubuntu Minimal "Live" USB Drive, just like Kali.

In case of Kali, it was possible to build minimal live USB with persistence using kali-linux-light-2019.1-amd64.iso. The light version has far fewer packages. The idea is to run a lighter resource demanding environment whereby you can download the tools individually or by groups at a time as you need them. Of course, you could go right ahead and download them all and convert light to full.

In case of lubuntu it is not possible because Ubuntu light ISO builds are not available. Probably because there's no much advantage of maintaining such builds. The kali project also discontinued maintaining such separate light ISO builds.

The solution to build live USB with Minimal Lubuntu and persistence can be found here - Linux From Scratch. It will involve building custom build of ubuntu ISO then putting it on USB using Rufus with persistence enabled. Here's the step-by-step article to follow after we build light ISO.

PS: Answer is extracted from the discussion on comment thread with @user535733 and @karel