The residual life of a random variable X at random time Θ is defined to be a random variable XΘ having the same distribution as the conditional distribution of X − Θ given X>Θ(denoted by XΘ = (X − Θ|X> Θ)). Let (X, Θ1) and (Y, Θ2) be two pairs of jointly distributed random variables, where X and Θ1 (and, Y and Θ2) are not necessarily independent. In this paper, we compare random variables XΘ1 and YΘ2 by providing sufficient conditions under which XΘ1 and YΘ2 are stochastically ordered with respect to various stochastic orderings. These comparisons have been made with respect to hazard rate, likelihood ratio and mean residual life orders. We also study various ageing properties of random variable XΘ1. By considering this generalized model, we generalize and unify several results in the literature on stochastic properties of residual lifetimes at random times. Some examples to illustrate the application of the results derived in the paper are also presented. © Brazilian Statistical Association, 2018.