  First we give conditions on a Banach lattice $E$ so that an operator $T$ from
$E$ to any Banach space is disjoint $p$-convergent if and only if $T$ is almost
Dunford-Pettis. Then we study when adjoints of positive operators between
Banach lattices are disjoint $p$-convergent. For instance, we prove that the
following conditions are equivalent for all Banach lattices $E$ and $F$: (i) A
positive operator $T \colon E \to F$ is almost weak $p$-convergent if and only
if $T^*$ is disjoint $p$-convergent; (ii) $E^*$ has order continuous norm or
$F^*$ has the positive Schur property of order $p$. Very recent results are
improved, examples are given and applications of the main results are provided.
