Many developers are reporting that they have been unsuccessful in getting the initial beta-release of macOS 10.14 Mojave to install on external drives. In many cases, they are connecting the external drive via a USB-C adaptor to a MacBook or MacBook Pro.
I too had exactly this problem: I connected two different USB3 SSDs to my MacBook Pro 14,1 using a good quality USB-C to USB adaptor. Each time that I ran the Mojave installer to install on that external SSD, all it did was copy the install files into a folder, restart, then drop me back into High Sierra from the Mac’s internal SSD.
Others have reported the install aborting at various other points, sometimes late in the installation. Some have suggested running the installer from the command line rather than the Finder.
My solution was to connect the external SSD via a CalDigit TS3 Thunderbolt 3 dock. If you haven’t got a similar dock available, but your external drive has a Thunderbolt 1 or 2 port, try connecting it direct using a Thunderbolt 3 to 2 adaptor.
Once you have got the Mojave beta installed on your external drive, you may be able to revert to using the more basic adaptor.