New Delhi, Sept. 3 -- As HDFC Bank scrambles to find a successor to chief executive officer Sashidhar Jagdishan, the country's largest private lender confronts a question that might bother its competitors too: do India's private banks have enough leadership depth to manage a new era of recurring CEO transitions?

Kotak Mahindra Bank is also looking for a new chief, and ICICI Bank faces a CEO transition next year. Private banks are going through more frequent leadership changes than earlier, when stalwarts including K.V. Kamath, Aditya Puri and Uday Kotak held the reins for decades.

What they have left behind is a private banking landscape marked by high key-person risk, a shallow pool of all-round banking leaders and reliance on familiar...