  Although the unavoidable minors of large 3-connected matroids were found
nearly thirty years ago, there has been little progress on solving the
corresponding problem for large 4-connected matroids. This paper aims to take a
step towards solving that problem. The objects of study here are 4-paths, that
is, sequences $(L, P_1, P_2, \ldots, P_n, R)$ of sets that partition the ground
set of a matroid so that the union of any proper initial segment of parts is
4-separating. Viewing the ends L and R as fixed, we call such a partition a
4-flexipath if $(L, Q_1, Q_2, \ldots, Q_n, R)$ is a 4-path for all permutations
$(Q_1, Q_2, \ldots, Q_n)$ of $(P_1, P_2, \ldots, P_n)$. A straightforward
simplification enables us to focus on $(4,c)$-flexipaths for some $c$ in
$\{1,2,3\}$, that is, those 4-flexipaths for which $\lambda(Q_i) = c$ and
$\lambda(Q_i \cup Q_j) > c$ for all distinct $i$ and $j$. Our main result is
that the only non-trivial case that arises here is when $c = 2$. In that case,
there are essentially only two possible dual pairs of $(4,c)$-flexipaths when
$n \geq 5$. A key technique in the proof of this result is of independent
interest. We construct the clonal core of a partitioned matroid. From the
relatively simple structure of this clonal core, we can deduce many properties
of the original partitioned matroid from the local connectivities between
unions of parts of the partition.
