So is the migration something that the app on the phone could do (sort of automatically/seamlessly)?
E.g. Breez wallet (@roy ;) ) could have a button in the settings - "Switch to provider ABC". You tap it, it asks "are you sure you want to migrate to another provider, this will take 5 minutes and 500 sats". 
And then it either does automatically send the sats from older provider to newer, or it could do something smarter - like downloading channel states, making old provider disabled and pushing those to new provider.

I'm just brainstorming - the main risk with greenlight-like setup is that the provider will block me from transacting (by e.g. going offline). So I'm trying to see what are the mitigations. 

nout

bitcoin

Stacker News Roundtable #2 - LSPs

That involves you exporting your exact channel states (commitment transactions) and taking downtime to switch providers. 

I think it will always be easier to just move your funds and setup anew with some other provider. You can use BIP32 to maybe reuse the same seed, but actually moving channels and keys doesn't make much sense to me. 