Step 1: Understanding the Concept:
Foreign travelers visited India during different dynastic periods, and their accounts provide a valuable external perspective on Indian history.
Step 2: Detailed Explanation:
1. Marco Polo (C): Visited the southern tip of India (Kayal) in the late $13^{th}$ century (c. 1292 CE) on his way back to Venice from China.
2. Ibn Battuta (D): Arrived in India during the $14^{th}$ century (1333 CE) and served as a Qazi under Muhammad bin Tughlaq.
3. Duarte Barbosa (A): A Portuguese official who traveled through the Vijayanagara Empire in the early $16^{th}$ century (c. 1500–1516 CE).
4. Francois Bernier (B): A French physician who was in India during the mid-$17^{th}$ century (1656–1668 CE), during the reigns of Shah Jahan and Aurangzeb.
Step 3: Final Answer:
The chronological order is: Marco Polo ($13^{th}$ c.) $\rightarrow$ Ibn Battuta ($14^{th}$ c.) $\rightarrow$ Duarte Barbosa ($16^{th}$ c.) $\rightarrow$ Francois Bernier ($17^{th}$ c.).
The sequence is C, D, A, B.