I have not got the book in front of me but I believe the ANO requires that, at night, an aircraft under tow should display those lights it is required to display in flight - i.e. Nav Lights and a red anti-collision light if fitted. During the day there is no requirement.
The practice of displaying strobes when crossing or entering a runway has developed I think as a 'best practice' measure following the famous incident at EGLL when an aircraft made a very low go-around with one lined up. (I think the aircraft on final could not see the one on the threshold.
Certainly, where I work, if you give an aircraft line up clearance, just before they enter the runway most will activate white strobes. One particular operator will also switch them on when crossing any intermediate runway.