Start with the date, not the paperwork
Before you change anything, settle what your status actually is for the year you moved, because every other step is dated from it and a wrong date makes the rest wrong too.
Settle it from your actual days in India under Section 6, then test whether you qualify as Resident but Not Ordinarily Resident under Section 6(6). That is what decides whether India taxes only your Indian income or your worldwide income. Get it wrong and you either overpay on foreign income India never had a claim on, or under-report income it did.
The full version, including how the year of the move is split: Your residential status in the year you move.
PAN, and the status the tax system reads
Your PAN record is what the banks and the tax system both read, so it comes before the money steps.
If you have taken a foreign passport or hold an OCI or PIO card and have never had a PAN, apply for a fresh one on Form 49AA, not the resident Form 49A, and get any foreign document apostilled or attested by an Indian embassy first. Without a PAN, tax on your Indian interest, rent or property sale is withheld at 20% under Section 206AA, and you cannot file a return, claim a refund or apply your treaty rate. See PAN on a foreign passport.
If you already have a PAN, put your non-resident status on record, either by filing a return as a non-resident or by intimating your jurisdictional assessing officer with proof. The Aadhaar-linking exemption is not applied to your PAN automatically. With nothing on record the system still reads you as a resident who did not link, so your PAN is flagged inoperative, tax is deducted at 20% or the rate in force, whichever is higher, and refunds can be held back. See PAN inoperative and the 20% deduction.
The tax portal, and which officer holds your file
On the e-filing portal open My Profile and change your standing residential status to non-resident with your overseas address current. Then ask the assessing officer who currently holds your PAN to migrate it to the international-taxation circle.
If it still reads resident, your PAN can be flagged inoperative for Aadhaar non-linking and tax cut at 20%, your case sits with a local officer instead of the international-taxation circle, and bank KYC or refunds stall because the records disagree.
The step-by-step, including the jurisdiction migration: Updating your residential status on the tax portal.
Bank accounts, in whichever direction you are moving
This is the step people leave longest and it is the one that freezes.
If you have become a non-resident: have the bank redesignate your old resident savings or current account as an NRO account from the date you became a non-resident, and open a fresh NRE for foreign earnings. Leave it and the bank's re-KYC picks up your non-resident status and can freeze the account or put it under review, while interest keeps being deducted at the resident TDS rate instead of the non-resident one. See Resident account still running after you became an NRI.
If you have moved back to India: tell your bank within about 30 days of becoming a person resident in India under FEMA. Redesignate the NRE and NRO accounts to resident accounts, open an RFC to hold the foreign-currency balances, and let FCNR deposits run to maturity before converting them. Leaving an NRE account quietly running is a FEMA breach, and its interest exemption under Section 10(4) ends the day you become resident under FEMA, so the interest becomes taxable even while you are still RNOR. See Redesignating your accounts when you return.
Demat and trading, which is not the same as the bank
The demat is a separate conversion and it is routinely missed, because people assume telling the bank covers it.
Convert to an NRI demat and trading account: move your resident-era shares to an NRO-linked demat by off-market transfer, which does not mean selling them, set up an NRE route for future repatriable investing, and close the resident demat.
Continuing to hold and trade on the resident demat is a FEMA contravention, and it means your gains have been reported and taxed as a resident's rather than an NRI's. The full conversion: Converting a resident demat and trading account.
If you have not left yet, do the whole list first
Every step above is easier before you go, because you still have Indian address proof, a working Indian phone and a branch you can walk into.
Before you leave, get your resident savings and current accounts redesignated to NRO, open an NRE (and FCNR if it suits you) for money you earn abroad, update your PAN and bank KYC to non-resident, and tell everyone who deducts tax on your Indian income, your bank, your tenant, your fund houses, so TDS is applied at non-resident rates.
Left until after you have gone, it turns into frozen accounts, wrong TDS and a residential status flip you cannot pin down, and running a resident account once you are a non-resident is a FEMA compliance breach. The pre-departure version: Before you leave India.