Whatever approach we use to Eastern, Western, Northern and Southern minds depends on that person, their background, their level.
Some would attempt to find Dharmakirti's proof very dry, boring and beyond what they could understand. Many Americans, Belgium people, and Italian people I have come across find various explanations helpful. There are times I attempt to assert more philosophical explanations, the doctors loved it, the housewives from the US I met tried to stay awake.
The subtle mind has no beginning and no end. At the end of our lives or during sleep for example the gross minds dissolves into the subtle mind. Hence in dreams we can 'see, feel, hear, etc' but it is without our gross aggregates..it is the mind seeing 'itself' in the case of dreams, the dreams arise from the mind and are viewed by this mind. That is one proof the mind can exist without it's physical corporeal form. Even with that scientists wouldn't be fully convinced, not that it matters, because they assert it is brain activitiy or memory activity or combination.
Since the continuity of the mind is like a film reel that it is constant and each picture frame is depending on the preceding and following frame to form a picture, our minds are similar. Hence if we are existent today, we must have been existent yesterday, the day before, the month before, the year, decade and even up to our birth. Even at birth when the red cell of mother and white of the father joins and becomes our first form that is when consciousness enters. Hence, if the two cells joined at 11pm, and the mind started to exist then, that would be illogical as the mind just doesn't exist and appear suddenly. Hence at 10:59 pm we must have had a mind, at 9pm, 6pm, the day before, the month before there was a continuity of the mind that continued up to our conception at 11pm. Hence there is memory. If the continuity of the mind was cut, there could be no memory. Since it is continuous, there can be memory. Even things that we experienced at 2 or 3 years old can be retrieved by meditation, regression therapy or hypnosis. That shows it is stored. Hence memories of previous lives can be tapped into especially when we are very young. As the 'distance' from the previous life to the current is not that far. Many incarnate lamas can remember clearly their previous existences. Many children can also. As remembering previous lives are not the exclusive domain of tulkus.
Scientists/skeptics would have many explanations against that as we have heard such as cultural, environmental and even genetic imprinting, etc etc..
Similarily, we can fast forward. If we were to die at 8pm, our mind having had prior functioning existence, will logically continue its existence. So If we die at 9am, then for example sake, after the process of dissolving of the winds our minds have to continue even after the body has ceased to function. So at 9:01pm our minds would have to continue and then 10pm, the next day, the next month, the next year. Hence this existence prior to conception and this continued existence after death is where we slot in karma. Something has to control where we came from and where we go. I have written in brief, but when we studied this from our glorious teachers decades ago, it worked for me. But it didn't work for the majority. It was too sublte and abstract. In the audience there were mixed crowd but minority of a few Easterners. Over the years I have elaborated this point many times in my travels, most was not interested and wanted stories, testaments of high reincarnated lamas and what they can remember, and or combination.
Even among many confirmed Buddhists, there is a superficial belief in the above or reincarnation. Proof? All Buddhists would lessen dramatically their production of negative actions based on karma leading to reincarnation. Lowest of the scopes, fear of future lives' sufferings would make people refrain from accumulating further negative karma hence hold their refuge vows, bodhisattva and or tantric vows more dearly than their lives.
There are many ways to approach people of all backgrounds. Generalisations of how to approach various people due to geography works sometimes. Many times not.
I appreciate all approaches. It would require the pliability of the speaker to suit the various types of listeners.
"Using skilfull means drawn by the strong force of compassion,
May I clear the darkness from the minds of all beings,
With the points of the path as I have discerned them,
May I uphold Buddha's teachings for a very long time." ~Lam Rim Dedication Prayer
TK