  We formulate a notion of "punctual gluing" of $t$-structures and weight
structures. As our main application we show that the relative version of
Ayoub's $1$-motivic $t$-structure restricts to compact motives. We also
demonstrate the utility of punctual gluing by recovering several constructions
in literature. In particular we construct the weight structure on the category
of motivic sheaves over any base $X$ and we also construct the relative Artin
motive and the relative Picard motive of any variety $Y/X$.
