How should I stay in touch with employees of a client that recently left us?

It would be nice to stay in touch with people I actually worked/interacted with, so I was wondering if it would be frowned upon to stay in touch with them, or ask them for a reference, or maybe add them on LinkedIn?

I'll dissect and answer your question.

Yes, I can understand that it would be nice to stay in touch with people that you worked closely with, so staying in touch with them by itself shouldn't be frowned upon or look bad. You probably already have their contacts or phone numbers so you surely have a channel of communication with them.

Asking them for a reference? Well, the real question is do you want or need a reference right now? This one I would be careful as it can be interpreted as you were actively looking for a job, and in case you are not it could backfire.

Connecting/adding them in LinkedIn seems a perfectly fine and OK thing to do.

Finally, and a suggestion, maybe you want to send a "farewell message" to them where you thank them for their disposition and for the time spent together, the professional interactions you had, etc.. Maybe you can use that message to ask/suggest that you connect on LinkedIn if they desire to.

I am not a fan of LinkedIn unless people want to be contacted that way. I lost all respect for that system when it started asking my friends to confirm that I had skills I never claimed I had. That poisons its database, as far as I'm concerned, making it nearly valueless for its intended purpose.

What I would do is just send personal email to the folks whom I had been in touch with, saying that it had been a pleasure to work with them and inviting them to stay in touch. If you are actively looking for another position, mentioning that and asking them to let you know if they see anything go by that you might be interested in certainly won't do any harm, but putting the emphasis on the courtesy rather than the attempt at networking is likely to be received better.

Networking has been oversold, by the way. It can produce useful leads, sometimes, but it is more likely to do so if you stay in touch with friends rather than trying to manufacture connections with acquaintances.

Contact them all on the linkedin. Be friendly, "just keep in touch", and so on.

Note, even if they are not customer of your customer any more, your customer can give you a next project or not, on any reason or without a reason.

However, very likely it worths the risk.

Goal is, a possible sales strategy: regularly ping them. Just keep yourself in their mind.

For a while, do not give them direct offer. It would look as if you would take customers from your customer away. But later, that is fine.