How do I export my Google Fit data to iPhone and keep it in sync in the future? I have used Google Fit on my Android phone for a while, and recently switched to the iPhone. How do I import my step counts and other fitness data into Apple Health on the iPhone from Google Fit?

Are there cross-platform cloud services that store this data in the cloud and have both iOS and Android clients so that I don't have this problem again the next time I switch phones?

I posted the same question on apple.stackexchange.com to get answers from iOS users.
To answer the first part of your question - the Android part:

You export your Google Fit data by going to Google Takeout.  It is not exported from within the Fit app itself.

Takeout is part of Google's Data Liberation project.  It enables you to download a copy of your data held by Google and attached to your Google account.

The Fit data can be downloaded in TCX or CSV format.