I disagree with the premise of your question, but maybe I just do not understand it. When points collide (in the world of stable curves) a new component "bubbles off" - in other words, there is no difference between points colliding and the $\mathbb P^1$ breaking. So I think that if you want to understand degenerations of your curve in terms of what happens to the branch configuration "downstairs" then I think you should be looking at the theory of admissible covers.
If your 6-pointed $\mathbb P^1$ breaks into two pieces with three markings on each, then an admissible double cover branched at these points will have branching also at the node, so you get two elliptic curves glued at a point. If instead you have one piece with two markings and one with four, then the inverse image of the 2-pointed component is a $\mathbb P^1$ and the inverse image of the 4-pointed component is genus one curve, and these are now glued to each other at two points. This curve is not stable, but we can stabilize it, and we get an elliptic curve glued to itself.
Finally, what kind of explicit families are you looking for? Do you want to write them down in coordinates? And what is a "high-dimensional base" here -- a base of dimension more than 3 (= the dimension of the moduli space of genus three curves) seems a bit silly, right?