Short positive loxodromics in graph products

We give a method for effectively generating generalised loxodromics in subgroups of graph products, using positive words. This has several consequences for the growth of subsets of these groups. In particular, we show that graph products of groups with strong product set growth properties also share those properties. We additionally show that the set of growth rates of a class of subgroups of any graph product of equationally noetherian groups is well ordered.