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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u({\eta }_1,t),\cdots ,u({\eta }_q,t)$ with $0\le {\eta }_1<{\eta }_2<\cdots <{\eta }_q<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $\left({\mathrm{P}}_q\right)$ of (P) in which the nonlinear term contains the sum $S_q\left[u^2\right]\left(t\right)=q^{-1}{\sum }_{i=1}^q{u}^2(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $\left({\mathrm{P}}_q\right)$ converges to the solution of the corresponding...