Let $L_{m,c}$ stand for the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over a field $K$ of characteristic zero. Automorphisms of the form $\varphi(x_i)=e^{adu_i}(x_i)$ are called pointwise inner, where $e^{adu_i}$, is the inner automorphism induced by the element $u_i\in L_{m,c}$ for each $i=1,\ldots,m$. In the present study, we investigate the group structure of the group $\text{\rm PInn}(L_{m,c})$ of pointwise inner automorphisms of $L_{m,c}$ for low nilpotency classes.