We investigate a large class of random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions $din{1,2}$ we completely characterise recurrence vs transience of random walks on the infinite cluster. In $dgeq 3$ we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [P. Gracar et al., The age-dependent random connection model, Queueing Syst. {bf 93} (2019), no.~3-4, 309--331. MR4032928].
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