How can H1B transfer be done from India after second extension?

I am in India and my second H1B extension was approved after I returned to India. However my Indian employer has not handed me my petition documents since the petition was approved and papers were mailed to employer’s US address after I returned to India. Now I have the latest file number, original visa petition, first visa extension document, but don’t have the hard copy of latest(second) approved I797 (petition papers) with me. I am thinking to switch employers in India. However I want an opinion BASED on EXPERIENCE (not a simple asking the consultants, attorneys, etc) if the recruiting employer can use my approved petition to allow me work on their US assignment? Also, how would visa interview work in this case because my current visa has expired in passport but extension petition is approved?

Background - I worked in US for 1.5 years, returned to India for 7 months, and then worked for another 9 months.  Have 7+ years of experience in multiple roles.

Haven’t done it myself but I know it can be done. Once you have made through the cap (your 1st H-1 petition), you are cap-exempt for 6 years. You don’t need to necessarily have the most recent petition to file the cap-exempt petition but any of the approved petitions.

Saurabh, I understand that I am eligible for cap-exempt status. However I want to know how would the process work when you just have a file number of the approved petition without the approved petition documents?

When switching jobs in India, can the new recruiter use my existing petition using approved file number without latest approved petition papers? If YES, how much delay can I practically expect from USCIS given that all the filing is done from India? (data based on experience would be great)