For $X$ and $Y$ finite sequences of finite trees, let us say that $X$ is everywhere contained in $Y$ ($X\subseteq\_{ec}Y$) iff, for every $y\in Y$, there is some $x\in X$ such that $x$ is a minor of $y$.
Is $\subseteq\_{ec}$ a well-quasiordering on the set of finite sequences of finite trees? (I.e.: Does every infinite set of finite sequences of finite trees contain two elements $X$ and $Y$ such that $X\subseteq\_{ec}Y$?)
Kruskal's theorem for labelled trees implies that this holds when all sequences in the set have the same length (or are of bounded length).



A variant of Kruskal's theorem
To reformulate your question, given a wqo $(A,\leq)$, here finite trees with the minor ordering, you ask whether $({\mathcal P}\_\mathrm{fin}(A),\sqsubseteq)$ the set of finite subsets of $A$ with a `minoring' ordering is also a wqo: $X\sqsubseteq Y$ iff $\forall y\in Y.\exists x\in X.x\leq y$.


This is does not hold for arbitrary wqos: it fails for any wqo that contains an isomorphic copy of the *Rado structure*, see [Jančar (1999)](http://dx.doi.org/10.1016/S0020-0190(99)00149-0). The wqos for which it holds are called *$\omega^2$-wqos* in a classification by [Marcone (2001)](http://dx.doi.org/10.1023/A:1013952225669).


Coming back to your specific question about trees, [Nash-Williams (1965)](http://dx.doi.org/10.1017/S0305004100039062) shows that trees under homeomorphic embedding have the much stronger notion of a *better quasi order*.


