get the contact details of the settlement department for securities from the receiving bank (email, phone), and communicate them to your old bank. This makes it much much easier to transfer and will heavily speed up the process
That's pretty much it. Sign paperwork with new bank and get both back-offices talking to each other.
Please ensure the new bank applies your original purchase price to newly delivered securities and not the price on day they are delivered. Otherwise PnL statements︂ will all be wrong which will cause problems if CGT is important in your investment︃ strategy. Even if they confirm this will be done double check after they have been︄ DFP's to your account.
P.S I have done this maybe half a dozen times with︅ Swiss banks especially and its very easy process.
Yes it will. However on transfer forms they often ask you to enter purchase price or "Cost Basis" of the stocks your transferring. One could in theory erase their capital gains by putting in favorable values if account was not previously known to current tax man 😉. But this is not recommended.
P.S OP should also be aware that unless your transferring it Private Bank to Private bank ensure names and addresses match on both accounts. If transferring to discount brokers like IBKR and︀ Saxo they don't like any differences in name and addresses details between accounts. Their compliance︁ will flag it as third-party transfers and common sense does not apply sadly in such︂ situations.