Thanks for your answer! I just found my question had a typo. In its current form, the answer is in fact trivial. I will revise my question now. Sorry about the confusion!

The use35593's argument requires the constant $c_2$ to be proportional to $1/d(x,y)$. As a result, the upper bound explodes when $x$ tends to $y$. To provide the desired upper bound in the question, where the constants $c_1,c_2$ are independent of t, x, and y, new techniques are required.