Use devtools-on-devtools:
open devtools and switch its Dock side
in the menu to a detached (floating) window

in the now detached devtools press CtrlShifti or ⌘⌥i on MacOS,
which will open devtools-on-devtools in a new window
UI method:
- in this new window switch to
Application
tab, expand Local Storage
, then devtools://devtools
on the left
- double-click
breakpoints
value on the right and copypaste it
- Now do the same on the target computer and reopen the main devtools window afterwards.

Console method (especially useful if the value is too long):
- run
copy(localStorage.breakpoints)
in devtools-on-devtools console on the source computer to copy the value to clipboard
- run
localStorage.breakpoints=prompt()
on the target computer
(the prompt will appear in the main devtools window).
Console method to export everything:
- run
copy(JSON.stringify(localStorage))
in devtools-on-devtools console on the source computer to copy the value to clipboard
- run
Object.assign(localStorage, JSON.parse(prompt()))
on the target computer
(the prompt will appear in the main devtools window).
P.S. Next time you can quickly toggle the detached state of devtools by pressing CtrlShiftD