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This paper addresses the problem of estimating multiple quantiles from a time series. A flexible class of dynamic multiple quantile (DMQ) models is specified, ensuring that estimated quantiles do not cross and that extreme quantiles are estimated by exploiting information coming from all the regions of the underlying distribution. DMQ models encompass the baseline model by Catania and Luati (2019), based on two main ideas, i.e. quantile spacings and score-type updates, and provide extension along several directions. First, a specification which includes a heterogeneous reaction of the left and right tails to past observations is included. Secondly, the case where all quantiles react differently to past information is considered. Finally, cross-tail effects are modeled, so that information from the right tail influences the left tail, and vice versa. The last specification allows the model to effectively capture the leverage effect in financial returns. Estimation is carried out by means of two stage quasi maximum likelihood estimators (2SQMLE). In the first stage the quantile intercepts are estimated by employing a targeting scheme. The use of empirical targeting versus parametric targeting is discussed. Consistency and asymptotic normality of 2SQMLE are derived. As closed-form expressions for h-step-ahead predictive quantiles are available, empirical results are assessed also based on comparisons in terms of the news impact curve, the quantile response function and the quantile decay, defined in this paper as the decay over time of quantile levels after an extreme shock.