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I think it's better if I tell a short example:

I plan on going from country A to country B, and I find out that it is cheaper if I pass by country X.
However, the flights from A to X and then from X to B have to be booked separately with unrelated airlines 1 and 2.

Is it usually possible to ask airline 1 to automatically transfer my luggage on arrival to city X, to the plane managed by airline 2?

Or do I have to grab my luggage myself and go through customs and airport-security?

Unfortunately, there is no "usually" in this scenario. Luggage transfer is only possible if there is a baggage interlining agreement, and only available if there is a specific policy allowing it. For example, some airlines allow intra-alliance throuch-check on separate tickets, some only allow it on same airline, and some disallow it altogether.

It is always preferable to have both flights ticketed on the same PNR, which may not be possible by buying directly, but may be possible by using a travel agent. If both/multiple flights are on one PNR, this allows additional luggage transfer interlining opportunities that othewise wouldn't be possible.

Edit: By PNR, I'm referring to a specific reservation or Passenger Name Record referenced by a short alphanumeric code termed a record locator. Multiple, otherwise separate flights may be booked under one PNR, and in some cases this may provide additional protection and capabilities such as the ability to transfer luggage.

Speaking from personal experience, I would strongly warn against this.

I had once a ticket from Raleigh, NC to JFK with American Airlines, where, 2 and a half hours later I was to change over to Air France and continue on to Stuttgart via Paris.

The lady at the check-in in Raleigh happily recorded all my flight numbers so I wouldn't have to pick up the luggage in JFK (where I also had to change terminals).

I'm not 100% sure, where exactly things went wrong (see footnote below[1]), but I arrived in Stuttgart, whereas my bag didn't. Since I was coming back from a 3-week business trip and leaving for another week-long trip the next day, it was most unpleasant to end up with only two sets of clean underwear.

The bag was, nonetheless, delivered a couple of days later to my employer's address.

[1] - Perhaps 2.5 hours was too short to transfer the bag between terminals and airline in JFK. On the other hand, the flight to Paris went behind schedule and I had to wait until the next flight to Stuttgart in CDG. Amusingly, the last time I flew via Paris (this time from Stuttgart to Moscow, on a single ticket with Air France), my bag also got lost. So it's hard to decide, what was the real problem, the change in airline or the shortcoming of Air France and/or the nature of CDG.