The paper deals with finite basis problem for classes of partition semigroups considered as involutary semigroups. A number of interesting results are obtained. The technique relies on a subtle analysis of the structure of the semigroups under consideration as well as on the results of a previous paper by the same authors [J. Eur. Math. Soc. (JEMS) 14, No. 3, 937-969 (2012; Zbl 1261.20068)]. It is in particular proved that the following involutive semigroups are not finitely based: the full partition semigroup \(\mathfrak C_n\) for \(n\geq 2\), the Brauer semigroup \(\mathfrak B_n\) for \(n\geq 4\), the partial Brauer semigroup \(P\mathfrak B_n\) for \(n\geq 3\), the annular semigroup \(\mathfrak U_n\) for \(n\geq 4\), \(n\) even or a prime power. It is also proved that, with respect to the `skew' involution, for each \(n\geq 4\) the semigroups \(\mathfrak B_n\), \(\mathfrak U_n\) and also the Jones semigroup \(\mathfrak J_n\) are inherently nonfinitely based. In addition, it is shown that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups.