The Kharosthi Script was more or less contemporarily with the Brahmi script, appearing around the 3rd century BCE mainly in modern-day northern Pakistan and eastern Afghanistan, although some examples do occur in India. Like Brahmi, Kharosthi seemed to have been developed for Prakrit dialects (which was the common speech of everyday life as opposed to Sanskrit which was the liturgic language). For instance, the earliest example of Brahmi and Kharosthi did not have the dipthongs /ai/, /au/, and the vocalic /r/ and /l/, which existed in Sanskrit but not in Prakrit. In particular, Kharosthi seemed to be used primarily for the Prakrit dialect of Gandhari, the language of the ancient kingdom of Gandhara. The evidence for this is in the form of a diacritic mark that denotes a transformation of an intervocalic constant (sometimes from a stop to a fricative), which existed in Gandhari.
Structurally, the Kharosthi and the Brahmi are nearly identical. The "letters" in both represent a constant followed by the short vowel /a/ (we'll denote this a "C-a" sign). Both denote change in vowel by adding marks to a sign. Consonant clusters are formed in both system by juxtaposing two signs closely together, sometimes forming a ligature. There are some difference, though. For one, while Brahmi had different signs for different initial vowels, Kharosthi used the same marks that change vowels in C-a signs on the sign for initial /a/ to denote other initial vowels. Another difference is that while Brahmi differentiated long and short version of the same vowel, Kharosthi used the same sign for both.
Eventually the Kharosthi Script fell out of use by the 3rd or 4th century CE, and the descendent of Brahmi eventually took hold in the northwestern South Asian.
This is the basic Kharosthi script.
And an example of strokes added to indicate different vowels following the consonant /k/.