I apologize if this is a slightly vague question.
Let's say that I have yearly data containing the amount of births, amount of deaths, and the total population in each of the 6 regions throughout the world. Let's assume that this data is accurate.
Using only this information, is there a way that I could calculate the number of people moving from one region to another in a given year or would I need more information?
Here is a simplified example with 3 regions rather than 6.
Pop_1_2000= 1000 Pop_2_2000 = 2000 Pop_3_2000 = 1500 Birth_1_2000 = 50 Birth_2_2000 = 75 Birth_3_2000 = 105 Death_1_2000 = 75 Death_2_2000 = 80 Death_3_2000 = 100
So total population in this "world" would start at 4500(add Pop_i_2000 up) and then be 4,475 after this first year (add starting populations and then subtract the births and deaths) . The populations in each of the three countries shift to this though. Pop_1_2001 = 1200 Pop_2_2001 = 1800 Pop_3_2001 = 1475
Is there any way that I could (only using this information) find out how many people moved from country 1 to country 2, country 2 to country 1, country 3 to country 1, etc.?
You would need more information. From the starting populations, births, and deaths you can compute what the population would be in each region for each year in the absence of migration. The difference between this calculation and the actual population is the effect of migration. With six regions there are fifteen net migration routes. You have six equations, one from the population of each region, and fifteen unknowns, the net migration between each pair of regions. It is even worse because there is at least one dependence between the equations.
A simple example would be three regions with no births, deaths, or change in population. It is consistent with the data that there is no migration at all. It is also consistent that a given number of people move from 1 to 2, from 2 to 3, and from 3 to 1.