The classical theorem of Lucas states that the binomial polynomials, which form a basis for integer-valued polynomials, satisfy a congruence relation related to their integer parameters. We consider here three problems connected with this result in the setting of discrete valued structures. The first problem asks for the shapes of Lagrange-type interpolation polynomials which constitute a basis for integer-valued polynomials and satisfy the Lucas property; the result so obtained extends a 2001 result of Boulanger and Chabert. For the second problem, we show that the Carlitz polynomials, which form a basis for integer-valued polynomials in a function field, satisfy the Lucas property, and derive criteria guaranteeing that Carlitz-like polynomials, which constitute a basis for integer-valued polynomials, enjoy the Lucas property. The third problem is to find conditions on general polynomials which form a basis for integer-valued polynomials ensuring that they satisfy the Lucas property.