  We present the full decomposition of the momentum fraction carried by quarks
and gluons in the pion and the kaon. We employ three gauge ensembles generated
with $N_f=2+1+1$ Wilson twisted-mass clover-improved fermions at the physical
quark masses. For both mesons we perform a continuum extrapolation directly at
the physical pion mass, which allows us to determine for the first time the
momentum decomposition at the physical point. We find that the total momentum
fraction carried by quarks is 0.532(56) and 0.618(32) and by gluons 0.388(49)
and 0.408(61) in the pion and in the kaon, respectively, in the
$\overline{\mathrm{MS}}$ scheme and at the renormalization scale of 2 GeV.
  Having computed both the quark and gluon contributions in the continuum
limit, we find that the momentum sum is 0.926(68) for the pion and 1.046(90)
for the kaon, verifying the momentum sum rule.
