  Using the hierarchy of weakly null sequences introduced by Argyros,
Merkourakis, and Tsarpalias, we introduce two new families of operator classes.
The first family simultaneously generalizes the completely continuous operators
and the weak Banach-Saks operators. The second family generalizes the class
$\mathfrak{DP}$. We study the distinctness of these classes, and prove that
each class is an operator ideal. We also investigate the properties possessed
by each class, such as injectivity, surjectivity, and identification of the
dual class. We produce a number of examples, including the higher ordinal
Schreier and Baernstein spaces. We prove ordinal analogues of several known
results for Banach spaces with the Dunford-Pettis, hereditary Dunford-Pettis
property, and hereditary by quotients Dunford-Pettis property. For example, we
prove that for any $0\leqslant \xi, \zeta<\omega_1$, a Banach space $X$ has the
hereditary $\omega^\xi, \omega^\zeta$-Dunford Pettis property if and only if
every seminormalized, weakly null sequence either has a subsequence which is an
$\ell_1^{\omega^\xi}$-spreading model or a $c_0^{\omega^\zeta}$-spreading
model.
