Some points to consider:
1. If you insure with a company and build up a no claims history no other company has to offer the same discount if you transfer. They probably will due to market forces, but they do not have to
2. If you obtain a cheaper quote on renewal, tell your original company before transferring cover. The first company will probably match the quote. So it always pays to shop around
3. If the car is insured under someone else's name (eg parent's) but you are the main driver, tell the company. It will probably not cost much more but not telling them may negate a future claim
4. Any no claims discount built up under 3. above will only be honoured by that company. They will not give you a letter confirming that you have built up a no claims history (because it is not your policy). Such a letter would be required by a new company.
Hope this helps.