These emigrants were to be settled at Albany, previously known as the Zuurveld. Described by Lord Charles Somerset as a ‘verdant carpet’ with fertile soil suitable for cultivation, cattle and pasture, 90 000 would-be settlers are said to have made application for their place in this paradise. The statistic is probably exaggerated but does indicate both the overwhelming response and the poor economic conditions in Britain at the time.
The actual number of people to take part in the scheme was eventually about 4000, divided between 60 parties and arriving on 21 ships between 6 April and the end of June 1820. Bitter disappointment followed: crops failed and the frontier remained turbulent. Some settlers managed to adjust to their new environment, but others left their allotments to seek employment in the towns. Whatever their trials, the immigrants' legacy to the Colony was deep and enduring.
If your ancestor was among the 1820 Settlers there is a considerable amount of published literature as well as information easily accessible online. It may therefore be unnecessary – or impractical, especially for family historians at a distance from SA repositories – to have recourse to original records, though these do exist. At Cape Town Archives Repository are Permissions Granted to British Settlers 1820-1824 (CO 6056 vol 2) and a list of immigrants in the year 1820 (CO 6137-6138); also Letters Received from immigrants 1820-1825 (CO 136,158,178,201,223,249).
Correspondence from leaders of the settler parties and others tell of setbacks which began even before embarkation. It was a major undertaking for any family to face a long voyage, in many cases with infant children, and the daunting prospect of a strange destination far from everything they knew. No wonder some of them were indecisive, getting cold feet before their departure, or experiencing unforeseen personal problems which prevented them from embarking as planned. One of my Gadsdens (spelled Gadsdon in this instance) appears on a published roll of the 1820 Settlers but as far as I can discover he never actually left England.
Footnote: there's some confusion as to the use of the terms 'emigrant' and 'immigrant'. One way to remember the distinction is to think of an emigrant as a person exiting from a country while an immigrant is an in-migrant, coming into a country. Each emigrant leaving his home country becomes an immigrant when he lands at his destination.