Suppose $M$ is a smooth manifold (compact if desired) with a cell structure or other nice stratification.
Call a path $\gamma : [0,1] \to M$ transverse to the stratification if there is a finite sequence $0 < t_1 < \cdots < t_k < 1$ such that for each $t\ne t_i$, $\gamma(t)$ is in the interior of a top dimensional cell, and for each $i$, $\gamma(t_i)$ is in the interior of a codimension-1 cell.
I’m not sure what is the correct notion of transverse for a path homotopy, but it should be something like this: using $s$ as the homotopy parameter, $\gamma_s(t)$ is in the interior of cells of codimension $c\leq 2$ for all $s$ and $t$, of codimension $c\leq 1$ for all but finitely-many pairs $(s,t)$, and of codimension $0$ for all but finitely-many $t$ for each $s$.
(So at most finitely-many of the paths $\gamma_s$ are transverse paths, and the exceptional $\gamma_s$'s only pass through codimension-$2$ cell interiors, and only finitely-many times.)
I guess this requirement could be made nicer since I'm not excluding things like a path touching a wall without passing through it.
Anyway, when are the following true?
Every path $\gamma : [0,1] \to M$ is homotopic to a path transverse to the stratification.
Every homotopy between transverse paths is itself homotopic to a transverse homotopy.
Ideally the first condition leads to generators or an enumeration of $\pi_1(M)$ as cycles along the dual graph of the stratification, and the second condition gives relations.
But, I'm not sure what further niceness assumptions are needed regarding, for example, how many top-dimensional cells can meet along a given codimension-1 or 2 cell.
Examples: $\overline{M_{0,n}}(\mathbb{R})$ and the cactus group; $\mathrm{Conf}_n(\mathbb{R}^2)$ and the braid group.
