While preparing for my group's first 5e game I was looking at the different classes and stumbled on a question I couldn't answer. For this example I will use druid as the class that the monk multi-classes to, but I suppose touch based cantrips from wizard/sorceror would also suffice.
A druid has access to the following Cantrip:
Cast time: 1 standard action
Effect: Flame in your palm
Duration: 1 min./level
Spell Resistance: Yes
Flames as bright as a torch appear in your open hand. The flames harm neither you nor your equipment. [...] You can strike an opponent with a melee touch attack, dealing fire damage equal to 1d6 + 1 point per caster level (maximum of 5) [...] No sooner do you hurl the flames than a new set appears in your hand. Each attack you make reduces the remaining duration by 1 minute. If an attack reduces the remaining duration to 0 minutes or less, the spell ends after the attack resolves.
To my knowledge (which could be wrong), cantrips level via character level as opposed to class level, so the switch away from druid would not effect the spell growth while building the monk levels.
So, my question is: If a monk were to cast this cantrip pre- or during battle, would their unarmed strikes receive the 1d6+x bonus for each strike as long as the spell had the duration to continue?
I was able to find an answer for the 3.5e rules (Can a monk's unarmed strike be considered as a touch attack?) however I was unable to find a dnd-5e equivalent.