For EOL products (End of Life), I know that companies stop selling them, but do they stop supporting them immediately?

NO ONE LIKES RISKS. THERE IS RISK MANAGEMENT THEORY THAT WILL SOLVE THE PROBLEM AUTOMATICAY. NO PROBLEM AT ALL.

No they don't do immediately, Mostly for the tech related products they will support for spares and after sales services but when the that products reach a timeline which was designed last for then they will gradually stop. 

EOL, end of life means they are out of support, while EOS end of sale, means they will not produce any more the product, but might have some in Stock, and usually after announcing eos there is a 5 year period till end of life, depends on the vendor and the product line

NO. companies dont stop supporting their EOL products instantly.for example take winows.when a new version of OS is available microsoft still provide support to its prior versions by releasing patches security updates etc.like wise take example of automobile industries.you will still get the spare parts available of a particular automobile variant even if its shut down from manufacturing.companies intimate their valued customer before annoucing the expiration of a particular product provided with the end of support date.(example windows-xp) support is not provided by microsoft now.

ahmmm for me? well let me tell you something. as my expirience in sales industry. we do still selling the EOL product until all of the new collections is arrive. the reason why we still selling the ol product is just to be added for sales and we promote also the new collections product. that time we can suggest to the customer the new products and that is the time to our nre products when its arrived. we give due date for selling the new products and that we called " LAst chance to by" we give 1 week due date to sell the old product and to be added on sales qoutas. after that we fulling out and promote the new products. that one of my expirience in sales. thank you

Normally that companies would treat EOL products by two ways, one way - to continue providing after-sales service such as build up necessary buffer for concerned old model parts repair/defect replace, eventually it will take on around half a year till absolute ending, another way - to promote new model immediately so that all clients can start to accept it till the old model phased out thoroughly.