Weighted extremal metrics on blowups

We show that if a compact K\"ahler manifold admits a weighted extremal metric for the action of a torus, so too does its blowup at a relatively stable point that is fixed by both the torus action and the extremal field. This generalises previous results on extremal metrics by Arezzo--Pacard--Singer and Sz\'ekelyhidi to many other canonical metrics, including extremal Sasaki metrics, deformations of K\"ahler--Ricci solitons and $\mu$-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.