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We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given by yn=(p+ryn-s)/(q+ϕn(yn-1,yn-2,…,yn-m)+yn-s), n∈ℕ0, with p≥0,r,q>0,s,m∈ℕ and positive initial values, and present some sufficient conditions for the parameters and maps ϕn:(ℝ+)m→ℝ+,n∈ℕ0, under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al. (2000), Iričanin (2007), and Stević (vol. 33, no. 12, pages 1767–1774, 2002; vol. 6, no. 3, pages 405–414, 2002; vol. 9, no. 4, pages 583–593, 2005). Besides, several examples and open problems are presented in the end.