can refer to

https://people.hofstra.edu/geotrans/eng/ch7en/conc7en/ch7c1en.html

Unlocking the Power of Urban Transport Systems for Better

Growth and a Better Climate

I think  that an improvement of transport infrastructure works in several directions: a) it increases utility of citizens saving their money and time on transportation between their houses and place of work, b) it makes this city more attractive for new migrants and commuters from rural neighborhood. An equilibrium city size is when city attractiveness is balanced by congestion effects. Hence, growing attractiveness of one city will cause migration to it from other areas.

If we focus on scale economies, the growth of population will stimulate economic growth as well. But if a city does not have technology or production responsible for economic growth, higher size will only increase pollution and make the citizens worse off. I think that social inequality dynamics depends on policy, but in non-controlled capitalism (free market) the social inequality in a city is typically increasing with growth.

As for an optimal city size and the role of scale economies, you can look at section 3 in my article: https://www.researchgate.net/publication/305307665_Population_Structures_in_Russia_Optimality_and_Dependence_on_Parameters_of_Global_Evolution

Article Population Structures in Russia: Optimality and Dependence o...

Maybe this publication as weel as other publications of the author will help you:

https://www.researchgate.net/publication/287952115_A_theoretical_assessment_of_the_EU%27s_smart_sustainable_and_inclusive_growth_policy_on_resource_use

Best regards,

Diana

Article A theoretical assessment of the EU’s smart, sustainable and ...

Improvement in transport infrastructure is expected to reduce economic/social inequality within a city. Thus, the two variables have inverse (negative) relationship.

To introduce a somewhat contrary view... but expanding upon Yuri's comment :

 "I think that social inequality dynamics depends on policy, but in non-controlled capitalism (free market) the social inequality in a city is typically increasing with growth."

Consider that we now know that development of 'free market' economies is accompanied by income polarisation, and that a significant contributor to this effect is the encouragement of 'flexible labour" policies.  These tend to impose lower rates of pay and longer working hours on the least skilled workers of societies, higher pay and shorter hours for those with 'key skills' and longer hours and considerable wealth upon the owning' or 'rentier' class.  

In the large metropolitan context this can result in each of the views expressed to be true, but only in part. 

The skilled 'middle class' that are required to work in the centre and at significant commercial nodes around it, but who aspire to live in less crowded circumstances, have the time to commute and the income to pay for it... they also have the leverage to secure wage premiums which are largely spent on an optimal mix of commute expenditure, housing cost and commute time.  They are the urban planners' 'model citizen'.

The rentiers are not only able to live and work in the central parts of cities but are also able to sustain, if they so wish, second 'weekend' homes far beyond the citiy to which they can commute quickly from the centres... often by air and high speed rail combinations.

The citizens providing the city's other two 'classes' with basic infrastructural services, upon which the city's very viability and hence their occupations depend,  do not have the free time nor the income to commute.  Nor do they have the income to afford 'standard' accommodation in the city centres, and commercial nodes, where their services are most needed... often at night.  They, and those under-employed and unemployed whose existence, as a reservoir of surplus resource, is required to maintain wages at the low levels required so that the skilled do not make even larger 'city' premium demands,  make up the inner city ghetto populations and the homeless who tend to migrate around the 're-development locations' of the inner cities and commercial nodes as property stock passes through their life-cycles from smart, to passe, to tired, to worn, to neglected, to decaying, to bulldozed.

So this picture is of 'economic growth' at the level of its social consequences and its spatial requirements. 

Since 'growth' when measured as GDP ( or more accurately GNI) is no more than the aggregation of generated profits and is inversely related to universal levels of well-being, measured at the individual level of longitudinal life experience and perception, and since wages optimisation of both the skilled and unskilled and also any public costs of urban transport infrastructures to move them between their places of residence and the workplaces are both costs against the maximisation of profit and hence are also negative to 'growth' the actual relationships are clear.

This, despite the flowery words and hopeful platitudes of 'city fathers' and property developers are the long-term effects and mechanisms you seek.  The policy implications are also clear... depending totally, of course, on one's particular philosophy of political economy... are you and the colleagues under whose judgement your work sits " All for one..OR.. One for all"?  Be careful in your choices...

An interesting way to look at this is to examine cities as they experience transportation investments/improvements. Many examples throughout history --- early ones would be horse drawn tram lines that opened new land for development. Later we have urban radial routes opening new rays emanating from a core city. Still later we have urban beltways with circular ring roads and connections between suburbs. And now, dense large cities often have new subway lines opening with accompanying density of new residential towers. All of these examples in some way show the role of transport in development -- there are opportunities opened up. Early adopters probably gain the highest initial advantages. Some of the transport improvement benefits are capitalized into the price of the land and rents will go up in the vicinity of places that are favored by new accessibility. Your question is how will this impact different socioeconomic groups -- some group will move to take advantage of the new open opportunities (the US phrase "drive 'til you qualify" means you go out as far as needed to reach affordable locations).

Many of these considerations can be turned upside down if the prime locations are central and become too expensive for some residents. Interestingly, urban settlement in some cities like Rio have very dense (poor) communities on highly accessible land -- the trade off is that it is very hilly and not at all well served by urban utilities. Other cities have such high density that transport becomes extremely congested. It seems like a great topic for a really careful comparative analysis.

very interesting to look at the differential impact of road/rail infrastructure on spatial distribution of population as a result of migrations, income distribution and incidences of urban/rural poverty. While on this, focus must be on the environmental impact of these differentiated spatial impacts.