There are several things that can run down the Towed's battery if you tow long distances with out driving the towed a bit
Brakes.. Many folks use Brake Buddy or one of the competitors which draw power from the towed and in all cases the brake lights may still draw power from the towed if you have aux brakes,,, and you should have aux brakes.
And of course parasitic loads that are always there
NOW: a New battery in good condition, LIkely no problem but as the battery nears end of life... problems.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times