Bootable USB stick might be the least complicated way, assuming the BIOS update .exe is a "universal" format file.

Copy the update .exe onto the bootable stick and then boot from USB...

sudo su

fwupdmgr refresh

fwupdmgr get-updates

fwupdmgr update -v

or download the firmware update from dell .exe file and put into /boot/efi

reboot press F12 ans choose firmare upgrade

root@ubuntu:/boot/efi# ls
EFI XPS_9370_1.7.3.exe

jensoroger.wordpress.com/2018/07/12/firmware-upgrade-dell-xps-13-kali/

Thank you - the laptop's firmware now is updated to the latest version.

Any hint on how I get the firmware of the TB 16 docking station updated?

Is the KVM based Windows 10 virtual machine viable?

Thank you!

Does the same work for non EFI boot partition? I run BIOS mode and only see /boot/grub. 

I have the same question. I can update the laptop but how do I update the Dock from linux.

Same here.
How could I upload TB16 firmware from Linux?

Most all firmware updates are Windows based so you would have to install them Using Windows.

There are not linux equivalents for Thunderbolt 3 security.

Put the BIOS update file on a bootable USB stick...?

Thunderbolt FIRMWARE requires actual windows.  Does not run on DOS.

This is because thunderbolt 3 uses signed security certificates.

After applying the upgrade (on Ubuntu), it asks for a reboot. Then, I get this warning "No bootable devices were found!Possible causes could be a corrupt OS image or a boot device not enabled in BIOS setup."

I reboot again and the laptop boots without any issue, but I can see the upgrade is still pending to be applied.

Any suggestion?