We explore economic segregation, social (ethnic) segregation, and long-term economic decline of neighbourhoods in cities as dynamic processes that emerge over time due to interactions between individual agents. Using the classic Schelling model as the basis of our work, we develop agent-based computational models to study both the emergence of such inequitable outcomes and also potential mechanisms that would enable reversing them. We find the emergence of wealth-based segregation, just as predicted by the Schelling model. However, our work posits that we can see a dramatic, non-linear decline in wealth segregation when even a small fraction of households move into neighbourhoods they cannot afford. The value of this 'small fraction' required for the onset of sharp desegregation decreases with the increase in the number of agents willing to move into less wealthier neighbourhoods. When the number of such willing agents is small, we find that over long stretches of time, more than a third of the richest and poorest neighbourhoods tend to retain their economic status compared to a much lower fraction of middle income neighbourhoods, corresponding to empirical observations from American cities. Therefore, for around one in three poor neighbourhoods, poverty can be a long-term reality. However, when we consider all neighbourhoods, change in economic status over time is relatively common, confirming the underlying intuition of descriptive anthropologies about neighbourhood change. We also find that enabling agent movement into neighbourhoods they cannot afford, precipitating non-linear transformation to desegregation, also increases the probability that poor neighbourhoods break out of their poverty traps. Finally, when agent movement in a city is mediated both by wealth and ethnicity, we find that decreasing wealth-based segregation has the effect of necessarily increasing ethnic segregation. This trade-off potentially enables the reconciliation of the empirical evidence on socioeconomic impacts of wealth and ethnic segregations.