With the Confederation of Canada in 1867, British North America had become the first dominion. Dominions were generally White settler colonies whom the government in Whitehall had enough trust in to govern broadly sensibly. The Australian Constitutions Act 1850 outlined the conditions through which the four Australian colonies and New Zealand would be awarded the right to 'responsible government'. By 1856, the five colonies of Australia (Victoria was split from New South Wales in 1851) and New Zealand had achieved responsible government. South Africa and Newfoundland, too, were to become dominions in 1907 and 1910. Australia became a Dominion in 1901, New Zealand six years later in 1907.

Andrew Heard is a constitutional scholar who briefly summarised how little legal control Canada had over its own affairs when it became a dominion in 1867 in this short paragraph:

At its inception in 1867, Canada's colonial status was marked by political and legal subjugation to British Imperial supremacy in all aspects of government—legislative, judicial, and executive. The Imperial Parliament at Westminster could legislate on any matter to do with Canada and could override any local legislation, the final court of appeal for Canadian litigation lay with the Judicial Committee of the Privy Council in London, the Governor General had a substantive role as a representative of the British government, and ultimate executive power was vested in the British Monarch—who was advised only by British ministers in its exercise. Canada's independence came about as each of these sub-ordinations was eventually removed.

Britain rarely invoked parliamentary supremacy in Canadian, or other dominions', affairs but strictly the reserve of the Westminster government for many decades was foreign relations. This is not too dissimilar to the arrangement in the devolved assemblies within the United Kingdom today, where Scotland, for instance, has competencies in most domestic issues, but strictly none on defence or foreign affairs. Dominions were forbidden to have diplomatic missions in foreign countries. All of these matters affecting citizens in Dominions would be processed in British embassies and consulates. Only in the 1930s and 1940s did Dominions begin to establish their own diplomatic missions.

An interesting scenario which seen some power and influence extended to the Dominions was the Great War. Colonial contributions to the British war effort fomented the creation of the Imperial War Cabinet in 1917. This allowed them to be involved in all manners of war decisions. Crucially, the Dominions signed the Treaty of Versailles in their own right and would come to become fully-fledged members of the League of Nations.

The decisions of the Balfour Declaration (1926) and the 1931 Statute of Westminster to grant greater (then full) autonomy to the British dominions was a conscious political decision of the Westminster government. Empire was unaffordable and allowing dominions to govern themselves with almost complete sovereignty, under the British crown for any necessary defence was a cost-effective way to preserve Imperial interests. It was hoped too, that Britain would benefit from keeping dominions at a distance, yet close enough to be able to use Dominion soldiers for the defence of the Empire. And there was a brief time when the Dominions, emboldened by their new-found sovereignty, publicly aired their reluctance to fight in another one of Britain's wars. It produced a particularly spicy meeting at the Imperial conference when South Africa's Prime Minister alluded to this idea. Alas, this eventuality wasn't to occur, and indeed, General Hertzog's neutrality would've been a contributing factor to losing his next election.

In summation, yes, Britain technically did have a lot of power in constitutional terms over its subjects in Dominions. However, it rarely had little interest in using that power, and over time, the sovereignty of Dominions increased and the popularity of the idea of Britain overruling a Dominion government within the Dominion itself became incrementally less.

Source:

Andrew Heard (1990): Canadian Independence