The great British missionary societies were founded within a generation of each other at the end of the eighteenth century, the Baptist Missionary Society in 1792, the London Missionary Society in 1795, the Church Missionary Society in 1799, alongside the older Anglican societies of 1698 and 1701. They were voluntary bodies raising money from congregations, not organs of government, and Andrew Porter's study is largely an argument against the assumption that they were simply the empire's chaplaincy.
The East India Company, which governed most of India until 1858, refused for decades to admit missionaries at all, on the plain commercial reasoning that interfering with Indian religion would cost it trade and peace. Missionaries were only permitted after evangelical campaigners forced a clause into the Company's charter renewal of 1813. Where missionaries did operate they frequently quarrelled with settlers, planters and officials, and in some colonies were regarded as agitators.
The clearest case of religion moving the empire rather than following it is abolition. The campaign that ended the British slave trade in 1807 and slavery in the empire in 1833 was organised largely by evangelicals and Quakers, argued in explicitly religious terms, and carried against powerful commercial interests. Its success became a lasting part of how the empire justified itself afterwards, including in the acquisition of new territory.