What, if anything, is the obligation of a new business owner to keep the promises to the people of the prior owner? Why?

None. If Obama wanted it to be permanent, he would have passed it properly through Congress not through an EO. EOs are reversed everytime the Pres party changes. 
Pres Trump is forcing Congress to do their job so these DACA recipients don't have to worry about it in the future. It is wrong to use them as a political game for the elections. That is why the Dems did not pass it as a law. 
So let's secure the border and pass DACA. 

A new business owner may honor the promises of the previous owner if he or she wants to keep the previous owner's customers.  Or they may go for cultivating entirely new customers. 

But the government is not a business (something Trump sometimes does not understand because he is so accustomed to a business environment).  But which promises of which president?   All presidents make promises and Trump seems to model himself (as far as he is able) on older presidents say of 40 - 50 years ago.  And receives support because many people regard those days as a kind of golden age in this country when people felt more hope. 

I frankly don't understand the complexities of that "DACA".  But I do feel resent presidents and politicians have gone overboard in coddling aliens who are here illegally.  These politicians apparently are unable to draw enough support from people born here so they resort to bringing people from other countries in and fund them on taxpayer money to get the needed votes.  Thus the whole "sanctuary city" movement.  They try to shape constituencies from immigrants to supplant those who have lived here for generations.  So they have learned nothing from the past themselves besides the craft and deceit of politics and vote-buying.