The real point of interest historically, is the two-way influence: how modernity or western culture may transform the way Buddhist teachings are understood or interpreted, evolving into new traditions altogether: but also, equally and maybe more significantly: how has Buddhism specifically influenced and transformed major elements of western culture and history in the interim. Probably both directions of influence need to be understood together because there must be some deep mutually interdependent relationship when looking with a wide lens.
This is really really important because, as we fail to remember sometimes, Western Imperialism transformed the entire planet. Values, mores, interpretations, philosophies, sciences, and worldviews from Spain, Portugal, France, England, the Netherlands, Germany totally transformed every culture with the few exceptions that locked them out (with more or less success) like Japan and Tibet.
So it's important to think precisely about that idea of modernism and how it's really influenced everything even before Westerners began to take an interest in Buddhism.
For example, people like the Navayana or Dalit (outcaste) Buddhists of India, who were reacting to centuries of caste-based oppression from Dharmic systems of hegemony and, as a result, converted to Buddhism (because it rejects caste) but then, with a strong influence from Anarchist, Communist, and other Modernist and Materialist thinkers, rejected Buddhist notions of reincarnation because they claimed to embrace Western Science and secular humanism (and reincarnation was the primary justification used for their status as Dalits) This movement has its roots in the late 1800s among a people who were, culturally, linguistically, historically closer to the origins of Buddhism than Tibetans or Chinese or of course any Westerners.