I am in the process of migrating from Ubuntu to Debian, and for logistical and health reasons the two laptops (Samsung and Asus) are miles apart and are not to be moved.

On the Samsung, I have Ubuntu 18.04 and Debian 10 co-existing on the hard drive.
Then I did a full install of Debian 10 on the Flash Drive and went through two steps:
1) Installing GRUB on the Samsung prevented me from accessing the Buster on the Flash Drive even after update-grub.
2) So, I went back, installed GRUB on the Flash Drive, update-grub, and lo and behold, I can access the three systems.

The drawbacks which I see (there may be more) are: a restart of either the Ubuntu or the Buster residing in the laptop does not work, only a shutdown will do; and that also means that I have to have the Flash Drive inserted at all times.

The situation is not all that bad from a security point of view: the Flash Drive is acting as a soft and hard key to grant access to the Samsung laptop. Not all that bad but not all that elegant either.

The Asus already has the Debian 10 freshly installed. To complete the strategy, I should insert the Flash Drive on the Asus and have a situation exactly parallel to the one working on the Samsung laptop. But before doing it, I would be grateful to hear your comments, no matter how unflattering , on how to go about this.

Thank you for your help.

Are all of the installs UEFI or Legacy? Or did you mix them? If they are EFI, on which drive is the EFI partition?

You have Ubuntu and Debian on the Samsung. Were they both functional? If so, what was the point of installing Grub again on the Samsung? If you had an install of Debian on a flash drive, booting the Samsung and running grub-mkconfig (or the Ubuntu stub update-grub) should have done the job.

are you getting a grub-rescue prompt on the samsung without the usb plugged in?

On the Samsung, the Buster Installer asked whether I wanted mandatory EFI, since it detected legacy stuff on sda1 and sda2. I responded No.
On the ASUS I configured the BIOS as Secure Boot 'disabled', and I read somewhere that EFI not mandatory in this case.

Yes, Ubuntu fully functional, Debian ok so far, but only sure after critical apps are tested.
Yes, after installing Debian on the Flash Drive, I did run update-grub from sda, but it did not pick up the Flash Drive Debian, and I did not know how to circumvent the problem except by doing as described.

(PS. Somewhere in AskUbuntu, I read the Ubuntu installer steps to create a full install in a USB Flash Drive, and noticed a crucial difference: whereas at the end the Ubuntu installer asked whether the drive is to be bootable or not, I did not get the same question when I worked with the Debian Installer)

If debian was installed in legacy mode on the asuse what you want to do should work on the asus also.
to better judge what is the ouput of and on the asus

Suddenly the Asus - a Zenbook 14 - has started giving problems; namely constant display flickering which makes it impossible to interact with cinnamon. Worse than that, now I can't even get a stable BIOS environment. Damn it.
ADDENDA: I managed to disable Fast Boot and the Asus SSD hard drive, and can finally boot from a Debian installer ISO, from where are available options for Graphical Install, InstalL, or GRUB command line.
How do you guys think I should proceed from here, bearing in mind that the display flickering has not gone away? Thks again.