I assume you are writing a computer implementation.
LAB is device independent. Your color codes are not, as they will be processed both by software and hardware before you see the colors. As a first approximation, convert your RGB to sRGB, which in this case will act as a reference device-independent color space.
When the sRGB values are obtained, convert to CIEXYZ and then to LAB, as the Wikipedia article on LAB suggests. Thus, write a function to compute $rgb\rightarrow sRGB \rightarrow XYZ \rightarrow L*a*b*$. Once the latter is obtained, you can interpolate linearly, e.g., take the average values of each value, and then perform the inverse transform to get your new RGB codes.