In the present paper, we introduce a family of the twice-iterated $$\Delta _{h}$$ -Appell sequences of polynomials based upon the discrete Appell convolution of the $$\Delta _{h}$$ -Appell sequence of polynomials $$Q_{n}(x)$$ . For these twice-iterated $$\Delta _{h}$$ -Appell polynomials, we prove an equivalence theorem and derive several determinantal properties in terms of the $$\Delta _{h}$$ -Appell polynomial sequence $$Q_{n}(x)$$ . We also find the recurrence relation, the shift operators and the difference equation satisfied by the twice-iterated $$\Delta _{h}$$ -Appell polynomial sequences. By appropriately specializing our results, we obtain the corresponding properties for the sequences of the twice-iterated Bernoulli polynomials of the second kind, the twice-iterated Boole polynomials, the twice-iterated Boole–Bernoulli polynomials of the second kind, the twice-iterated Poisson–Charlier–Bernoulli polynomials of the second kind and the twice-iterated Poisson–Charlier–Boole polynomials.