“I am wanting to know if we had allowed an easement across our property for another property owner to have access to their property behind us and now they are selling their property and the buyer is an adjoining property owner that will now have access to the property from his own property will be still have to allow access or use of the easement that is running through our property?”
It depends on the terms of the easement that you granted. Most access easements benefit the particular piece of property, not the specific owner. The legal terms are “appurtenant” (benefiting the property) and “in gross” (benefiting the person).
If you granted an easement for the property, the easement goes along with the property to the new owner, whoever that is. If you granted an easement to the prior owner as an individual, the easement goes along with that person. If you didn’t specify, it would be a matter of a court trying to interpret the parties’ intent, though for this sort of easement most likely it would be found to go with the property.
On the other hand, if you did not formally grant an easement, but only gave your neighbor permission to cross your property, you can probably withdraw that permission. You should, however, double-check the laws in your state on that point by consulting with a local attorney before you spring this on your new neighbor.