  Following the train of thought from our previous paper we revisit the
theorems of Pongsriiam and Termwuttipong by further developing their
characterization of certain property-preserving functions using the so-called
triangle triplets. We develop more general analogues of disjoint sum lemmas for
broader classes of metric-type spaces and we apply these to extend results of
Bors\`ik an Dobo\v{s} as well as those obtained by Khemaratchatakumthorn and
Pongsriiam. As a byproduct we obtain methods of generating non-trivial and
infinite strong $b$-metric spaces which are not metric.
