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Using latex makebst plus a bit of fiddling with the resulting .bst file, I've managed to get my bibliography close to the shape I want it to be in. The exception is for certain incollection and inproceedings fields. At the moment I'm getting entries like this:

Smith, John (2016), An article about things. In Joe Bloggs (ed.), A book about things, number 12 in A Series of Books. Publisher. City, 123–321.

What I want is for the entry to look like this:

Smith, John (2016), An article about things. In Joe Bloggs (ed.), A book about things. Number 12 in A Series of Books. Publisher. City. 123–321.

That is to say, in some locations I'm getting commas as separators where I want full stops. It's set up with natbib as the author-year support system and with \newblock between blocks, if that information helps.

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OK, it turns out that the part of the .bst file that I wanted to change is FUNCTION {output.nonnull}. Specifically, I've changed

output.state mid.sentence =
  { ", " * write$ }

to

output.state mid.sentence =
  { ". " * write$ }

This has caused some problems with capitalization elsewhere, but I have decided to hack through them at this point.

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