For a semigroup (Formula presented.) whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right- (Formula presented.)), the right diameter of (Formula presented.) is a parameter that expresses how ‘far apart’ elements of (Formula presented.) can be from each other, in a certain sense. To be more precise, for each finite generating set (Formula presented.) for the universal right congruence on (Formula presented.), we have a metric space (Formula presented.) where (Formula presented.) is the minimum length of derivations for (Formula presented.) as a consequence of pairs in (Formula presented.); the right diameter of (Formula presented.) with respect to (Formula presented.) is the diameter of this metric space. The right diameter of (Formula presented.) is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set (Formula presented.) has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on (Formula presented.), of all partial transformations on (Formula presented.), and of all full transformations on (Formula presented.), as well as the partition and partial Brauer monoids on (Formula presented.), have right diameter 1 and left diameter 1. The symmetric inverse monoid on (Formula presented.) has right diameter 2 and left diameter 2. The monoid of all injective mappings on (Formula presented.) has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on (Formula presented.)) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on (Formula presented.) has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.