Lipschitz interpolating sequences

Let $X$ be a metric space with a base point $0$, and let $\mathrm{Lip}_0(X)$ be the Banach space of all Lipschitz functions $f:X\longrightarrow \mathbb R$ such that $f(0)=0$. Given a set of points $\left((x_i,y_i)\right)_{i\in I}$ in $X^2$ with $x_i\neq y_i$ for all $i\in I$, we study the following interpolation problem: when for each bounded set $\left(\alpha_i\right)_{i\in I}$ in $\mathbb{R}$ the algorithm $$\frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\alpha_i\qquad (i\in I)$$ can be implemented by a function $f\in\mathrm{Lip}_0(X)$? Our approach involves the concept of a Beurling set of functions in $\mathrm{Lip}_0(X)$ for $\left((x_i,y_i)\right)_{i\in I}$ which has shown to be useful in the so-called transportation problem.