We study failure rate monotonicity and generalized convex transform stochastic order properties of random variables. We are especially interested in the effect of a tail weight iteration procedure to define distributions, which is equivalent to the characterization of moments of the residual lifetime at a given instant. The stochastic order thus defined describes a criterium to order lifetimes with respect to their ageing rates. We study the hereditary properties of both the monotonicity and order relation, and also address the how these generalized order relate with respect to other well known stochastic orders. Having in mind that the actual identification of an order relationship is computationally complex once we leave the exponential distribution world, we introduce a new criterium, based on the analysis of the sign variation of a suitable function. We use the criterium to establish order relationships either within the Gamma, within the Weibull families or between distributions suitably chosen among these two families. This criterium is also applied to derive ageing relationships among parallel systems formed with two components that have exponentially distributed lifetimes, showing that systems formed by equally distributed components age faster than systems with differently distributed exponential lifetimes. Moreover, we also show that parallel systems age faster as the number of components increases.