I don't currently use graph databases in production, but I do, however, have some experience.
I use both, in a similar vein of "using Elastic Search". It could be your primary store, but it's sometimes it's more pragmatic to have two sets, a "solid" base.
This is not to say that it can't be done. What I'm stressing is that larger "changes" are hard and difficult to handle - which means a lot in the start of your process, and less in the end, as in, when you're deciding how to model your data. For instance node layout (new properties? different type? other constraints?), and mass updates are also a bit cumbersome.
Usually I have more than one SQL table (naturally) since the data I've used in graph databases is mix and match (otherwise I'd just use a fixed schema and some relational DB).
-- As for "how that works", for me it's:
routinely update from my base database with queries alike: ID > last ID.
This has worked as expected, in terms of what data you get in, and which limitations you impose (e.g. timeliness).
I'm currently making a shift to running all data in my graph database as I've settled on a model (which edges, which nodes, which properties).
> querying from two different databases is going to slow down my responses
True, but depending on your data (do you know one of the queries beforehand - e.g. is your postgresql query enriching whatever your graph query returns) you might have success tying (inserting) some of the SQL data to your graph database.