A particular adaptation of Gale's top trading cycles procedure to school choice, the so-called TTC mechanism, has attracted much attention both in theory and practice due to its superior efficiency and incentive features. We discuss and introduce alternative adaptations of Gale's original procedure that can offer improvements over TTC in terms of equity along with various other distributional considerations. Instead of giving all the trading power to those students with the highest priority for a school, we argue for the distribution of the trading rights of all slots of each school among those who are entitled to a slot at that school, allowing them to trade in a thick market where additional constraints can be accommodated. We propose a particular mechanism of this kind, the Equitable Top Trading Cycles (ETTC) mechanism, which is also Pareto efficient and strategy-proof just like TTC and eliminates justified envy due to pairwise exchanges. Both in simulations and in the lab, ETTC generates significantly fewer number of justified envy situations than TTC.