How does core temperature relate with slow cooking?

For slow cooking you don't just want to achieve a specified core temperature. You want to hold the meat at that temperature (usually about 70 degrees Celsius / 160 Fahrenheit) for several hours until the connective tissue starts breaking down. The temperature of the oven / whatever is usually kept low (although it has to be at or above the desired core temperature, otherwise the meat will never come up in the correct temp) to prevent the meat from burning while it's cooked. Just bringing the internal temperature to 70 degrees will not produce pulled pork. Keeping it there for 2+ hours will.

You seem to be mixing up a lot of things here. For example, it makes no sense to say "Before achieving the core temperature, meat is rare which is not pulled." To avoid repeating information on the cooking of meat which has been written up on the site again and again, I will assume that you have read and understood Saj's canonical answer on roasts before continuing to read this answer.

Now on to details which are specific to your current question:

"Core temperature" is the temperature measured at a given point in the middle ("core") of the piece of meat. If your meat is in the freezer, its core temperature is -18 C. If your meat is in the slow cooker, its core temperature is constantly changing. So it makes no sense to say "before achieving the core temperature", the meat's core temperature is per definition the temperature it has achieved. (The information which temperature you want the meat to achieve is contained in Saj's answer linked above).

The way I parse the phrase "slow cooking temperature" is the temperature within the slow cooker. I am a bit unsure if you don't actually mean "the temperature which the meat should reach in order to count as slow-cooked" but since in slow-cooking, there is no such thing (see Saj's answer again), that definition would also render your question unanswerable. So I will stick with my first guess.

Having defined the terms, the answer to the question is: At the beginning of the cooking process, the slow-cooking temperature is higher than the core temperature. That is because the slow cooker heats up its interior first, but the meat's core takes time to warm up. Due to heat exchange between the meat and its environment, the meat gets warmer until it reaches the slow cooking temperature. From that point on, until the end of the cooking process, the core temperature is the same as the slow cooking temperature.

Generally speaking, "pulled" meat such as pulled pork or beef is made from large cuts of meat that have a lot of connective tissue and collagen in them, such as pork shoulder or ribs, beef chuck, etc. Lamb shoulder can often be cooked the same way but doesn't generally have as much connective tissue. Chicken thighs have some connective tissue but chicken breasts do not. The type and duration of cooking these all varies based on the type and size of the cut.

Also, generally speaking, connective tissue and collagen break down at around 160F and it takes a while for this to happen. Most "low and slow" recipes call for cooking at a temp somewhat above this, such as 200-250F, for many hours. The internal temp of the meat starts at room temp i.e. 70F and slowly increases until it hits 160F, and then usually "stalls" there, sometimes for hours (in the case of pork shoulder or ribs). During this time the connective tissue is bring broken down and released as gelatin and the meat becomes tender. If you keep cooking after this, the internal temp will go up to 170F, 180F and so on until it eventually gets to your cooking temp i.e. 200-250F, but usually by the time it's 180-190F it is very tender and falling apart and ready to be "pulled".

Many slow cooker recipes add liquid, this makes this a bit more complex since you're really braising or making a stew rather than simply cooking with dry heat. The same general things apply about the breakdown and temperature but you'll be left with a very different result. Also any fat that is in the meat will remain in the liquid which can make it unappealingly fatty or greasy. So you have to either be careful about how much fat is on the meat or have a step where you remove it later.

Almost all slow cooker recipes cook meat until it is very soft and falling apart, that is the main point of that type of cooking.