What I have understood so far:   You can slow cook to produce pulled pork/beef/lamb/chicken by applying relatively low temperature for long time so that proteins can denaturate (sous vide can help here to prevent drying out) Before achieving the core temperature, meat is rare which is not pulled.   So is slow cooking temperature (sure different for different proteins):   same or below core temperature, just for longer time, or somewhat beyond and is that estimated?  
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.  