For me its important that they speak British English - in daily life this means steering away from American media and watching cbeebies when they are little and pointing out if they use an Americanism that it is American - but it is easy, as we are in Germany, and the English taught in school is British English, they go to German school not international school, and we aren't really part of the ex-pat scene at all, as we live rurally.
Initially I thought I would want the kids to have English speaking friends, but although they know a few English speakers we see them rarely, and it hasn't seemed to matter. All the kids friends are German, which not only ensures their German is 100% native and they feel they belong here but also means their English is separate, a home language, which has worked out fine.
We speak English as a family language even though DH is German (he speaks pretty much perfect English, people often try to guess where he comes from and guess SA sometimes, but not that he is a non native speaker, so he doesn't "teach" them "bad" English), but now the kids are nearly 9, nearly 6 and 3 they do chop and change between German and English with DH - I always speak English to them, always, always, and I absolutely do insist they reply in English - but they have never resisted or complained, and very rarely reply in German, in which case I just ask them to say it in English, the same way I would remind them to say please (I believe most people avoid doing that, and would instead repeat back what they have said but in English, but I find that laborious, forced and totally unnecessary in our situation, unless it is vocabulary they might not have come across before because its a new topic of conversation - this only really comes up in relation to school subjects and materials).
I do also want the kids to be aware of things most British children would have as part of their lives, but I don't care at all if they identify as German - as indeed DC2 does, though DC1 identifies as British - or says she does -(we moved when she was 19 months, DC2 and 3 were born here).
The language and "Cultural Capital" matters to me - I have begun my plan to take each child to London the summer before they turn 7 as a 1 to 1 to do the main sites// museums, and plan to do so with each one again at early secondary age. We visit my family once a year - I know a lot of people do more, but its expensive and stressful! I make sure they know all the nursery rhymes, the myths and legends, and try with the popular culture too (through TV mainly).
The older 2 and I talk regularly about differences between their 2 countries, and try to be even handed - some things are better here, some there.
I agree with roots and wings 100% - but to be honest I want my children to feel rooted in the community they are growing up in - we have lived in the same tiny village for 7 years and don't have any plans to move until the youngest is through school (if we do move it would be locally, once they are all at secondary, maybe to a bigger town where the secondary schools are). Britishness is only one part of their identity and not their main source of roots. I feel I'd be doing them an injustice if they didn't have it though - especially the language at a proper native speaker level, as its a free gift they are entitled to, which can only help them in the future.