The present paper is concerned with a certain sequence of the nonlinear Durrmeyer operators $ND_n$, very recently introduced by the author [22] and [23], of the form \[ (ND_nf)(x)=ıt^1_0K_n(x,t,f(t))dt,\qquad\-eq xeq1,\quad nı\mathbb N \] acting on Lebesgue measurable functions defined on $[0,1]$, where \[ K_n(x,t,u)=F_n(x,t)H_n(u) \] satisfy some suitable assumptions. As a continuation of the very recent papers of the author [22] and [23], we estimate their pointwise convergence to functions $f$ and $\psi\circ|f|$ having derivatives are of bounded (Jordan) variation on the interval $[0,1]$. Here $\psi\circ|f|$ denotes the composition of the functions $\psi$ and $|f|$. The function $\psi\colon R^+_0\to R^+_0$ is continuous and concave with $\psi(0)=0$, $\psi(u)>0$ for $u>0$. This study can be considered as an extension of the related results dealing with the classical Durrmeyer operators.