I did a test on my a10, a year or so ago, and I wanted to see how much droop there was locally. my probe is right on the d1 5v line and you can see that as I twist the knob or use IR to get the relays to dance, they dance downward on the scope a little (lol). about 40mv of 'hit' when they get up and dance.
that was a clean, local, proper connection and that amount is not significant. BUT if you add wire and drive 4 boards from it (I was using only 1 d1 and 1 d2) you could definitely have voltage drop problems. you also hinted at this by saying that 5.3v was helping you get more relay action.
can you show an overhead shot of just the d1 board stack and its power wire and where that wire ends up? then a 2nd shot of what sends 5v into the lcduino.