How did decipherment of the Brahmi and Kharoshti script take place during the nineteenth century?

The decipherment of these scripts was undertaken by the British aided by the Indian Pandits. The most important man involved in this process was James Prinsep.

The process adopted for deciphering Brahmi script was:

1. The epigraphists studied several kinds of scripts written in Devanagari, Sanskrit, Prakrit, etc. They compared those scripts with the Brahmi scripts.
2. A thorough investigation of the Brahmi scripts led to the deciphering of the Asokan Brahmi in 1938.

The process adopted for deciphering the Kharoshti script was:

1. The coins of the Indo-Greek rulers helped in the deciphering of the Kharoshti script.
2. The coins contained the name of the Kings written in Greek and Kharoshti scripts. The name was compared with the letters of the script. This way a relation was established between the two and the script was finally deciphered.