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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 almostDunford-Pettis. Then we study when adjoints of positive operators betweenBanach lattices are disjoint $p$-convergent. For instance, we prove that thefollowing conditions are equivalent for all Banach lattices $E$ and $F$: (i) Apositive operator $T \colon E \to F$ is almost weak $p$-convergent if and onlyif $T^*$ is disjoint $p$-convergent; (ii) $E^*$ has order continuous norm or$F^*$ has the positive Schur property of order $p$. Very recent results areimproved, examples are given and applications of the main results are provided.

Disjoint $p$-convergent operators and their adjoints