I came across this problem in a proof of an existence result for stochastic differential equations with locally Lipschitz coefficients.
Let $f:\mathbb{R}^d\to \mathbb{R}^d$ satisfies the following: for any $r>0$ and any $x,y\in \mathbb{R}^d$ such that $|x|<r$, $|y|<r$, $$ |f(x)-f(y)|\leq C_r |x-y|, $$ holds, where the constant $C_r>0$ depends on $r$.
Now, take $\varphi\in C^\infty(\mathbb{R})$ such that $0\leq \varphi\leq 1$ and $\varphi(t)= \begin{cases} 1&\text{if } t\leq 1\\ 0&\text{if } t\geq 2 \end{cases} $. Let $$ f_R(x):=f(x)\phi(|x|-R). $$ Now the claim is for any $x,y\in \mathbb{R}^d$, the global Lipschitz condition $$ |f_R(x)-f_R(y)|\leq C |x-y|, $$ holds.
Here is my proof:
Case I: For $|x|\leq R+2$, and $|y|\leq R+2$, first note that we have $$ |f(y)|\leq |f(0)|+C_{R+2}|y|\leq |f(0)|+C_{R+2}(R+2)=:C'_R<\infty, $$ where $|f(0)|<\infty$ follows from $f$ being continuous on any closed and bounded set in $\mathbb{R}^d$. Hence, it follows \begin{align} |f_R(x)-f_R(y)|&\leq |\phi(|x|-R)||f(x)-f(y)|+|f(y)||\phi(|x|-R)-\phi(|y|-R)|\\ &\leq \max_{\mathbb{R}^d}|\phi|C_{R+2}|x-y|+C'_R\, C_\phi |x-y|, \end{align} where in the last step we used $\phi$ being global Lipschitz (as it is smooth and compactly supported).
Case II: For $|x|\geq R+2$ and $|y|\geq R+2$, we have $$ |f_R(x)-f_R(y)|=0\leq |x-y|. $$
Thus, using the argument in the accepted answer and comment of A step in proof of existence for SDE: handling locally Lipschitz function, we conclude that for $R>0$ fixed, $f_R$ is globally Lipschitz.