As Pat said, there's plenty written about the post-Famine period.
Irish Genealogy Toolkit has a brief overview of emigration, including migration to Britain. https://www.irish-genealogy-toolkit.com See topics "Irish immigration to England"
https://www.irish-genealogy-toolkit.com/Irish-immigration-to-England.ht…; See also pages " Irish immigration to Britain" and "Emigration from Ireland in the 19th century". A table shows English counties with the largest percentage of Irish-born population on 1851 census. Cheshire was 2nd highest at 5.7%. Lancashire was top with 8.9%. Lancashire was a popular destination for Mayo people. Main factors influencing choice of place were proximity to Ireland and employment opportunities. There were several ports on the north-west coast of England connecting sea-traffic to Ireland. The port of Liverpool was connected by rail to Manchester in 1830, the oldest inter-city railway in the world. Manchester to Macclesfield railway opened 1849. Trains had 3rd-class carriages for less well-off people. Cheshire had several manufacturing towns, plus mills in rural areas. The county has good farming land and a dairy industry (Cheshire Cheese) . There was work in mills for men, women and children. Some full-time farm work + seasonal.
There was a long tradition of seasonal migration to Britain for agricultural or construction work.
There are informative articles on many topics on this site too. Click on "News" at the top of this page and browse.
There were different patterns of migration.
Cheshire included Stockport, a town which had grown rapidly during the Industrial Revolution to 30,000 by 1851, manufacturing silk, then cotton and hats. Silk manufacture in Stockport and most other Cheshire towns stopped after 1850. Macclesfield then became the centre of Cheshire's silk industry and was known as "Silk Town". Macclesfield firms exhibited at the Great Exhibition 1851. Two Macclesfield silk weavers emigrated to New Jersey in 1839. They opened mills there, establishing the American silk industry in 1846. When times were hard, Macclesfield folk emigrated to work in the New Jersey silk mills. Fashions changed and Macclesfield silk manufacturing contracted in size. Population of the town declined by 3,000 between 1851 and 1861 due to depression in the silk industry. Paradise Mill, built mid-19thC, is a working museum. Photo of interior, showing looms https://www.macclesfield-live.co.uk/news/old-mill-heart-macclesfield-si… or do an internet search for Paradise Mill or Macclesfield silk museum.
Two other events during 1860s had a detrimental effect on Cheshire's prosperity.
The "Lancashire Cotton Famine" 1861-5 also affected Cheshire. It was caused in part by interruption of supply of raw cotton during the American Civil War. The cotton industry was a vital part of the British economy. There were 2,100 cotton mills in Lancashire and Cheshire in 1861. Approximately 60% of the spindles and looms were idle by October 1862. An estimated 50% of cotton operatives were unemployed; another 33% were on half-time; approximately 10% were fully employed. The huge increase in poverty caused other businesses e.g. shops to fail. There was a riot in Stalybridge and unrest in other Cheshire towns such as Dukinfield and Hyde. Chester Town Hall was set on fire.
Cattle plague 1865-6. Cheshire, an important dairying county, was affected worse than any other county in England. Cattle stocks were devastated. Healthy cattle were killed in an effort to halt the spread of the disease. The county was left in debt for the next 30 years. There were 2 other outbreaks of serious diseases of cattle in the decade, including foot-and-mouth.
Did the Moran family emigrate to Australia on an assisted passage?
Browsing census for Macclesfield from 1851 onwards, you may notice other Irish people from Mayo or perhaps recognize common Mayo surnames.