Posets of $\lambda$-compact sets

One directed partially ordered set $Q$ is a Tukey quotient of another, $P$, denoted $P \geq_T Q$, if there is a map $\phi : P\to Q$, called a Tukey quotient, that takes cofinal subsets of $P$ to cofinal subsets of $Q$. For a cardinal $\lambda$ a space is $\lambda$-compact if every open cover has a subcover of size $<\lambda$. For a space $X$, we consider the partially ordered set of all $\lambda$-compact subsets of $X$ ordered by set inclusion. We will study Tukey ordering and Tukey-invariant order properties of such posets where $X$ is a subset of a cardinal.