To move the active menu, hold down the move mod key (default Left Shift), hover over the active menu and hold down the move key (default Mouse Left) while dragging the menu.
To "detach" a menu, making it persistent, hold down the detach mod key (default Left Shift), hover over the active menu and press the detach key (default X). To reattach a menu, do the same to a detached menu when there is no active menu.
To close a detached menu, hold down the close mod key (default Left Shift), hover over the menu and press the close key (default Z).
You can let me know if it doesn't work right with some custom UI mod, but I doubt I can fix custom UI elements not moving with the default ones. It's more likely that custom UI mods can make themselves compatible with this menu by updating elements' positions based on the menu's position rather than expecting the menu to be stationary.
I've added compatibility with Chests Anywhere.
Technical
Requires SMAPI, uses Harmony.
Compatible with Mod Updater for automatic updates.
Code is at https://github.com/aedenthorn/StardewValleyMods.
If you want to complain or ask for help or help me test my mods, you can visit my Discord server.
A list of all my mods for Stardew Valley is available at https://www.nexusmods.com/stardewvalley/articles/895.