Line parametrized by ⟨t,t⟩

Animate[Show[Plot[x,{x,0,30},PlotStyle->Thick],Graphics[{PointSize[.02],Point[{t,t}]}]],{t,0,25},AnimationRunning->False,AnimationRepetitions->2]

Line parametrized by ⟨3 t, 3 t⟩

Animate[Show[Plot[x,{x,0,30},PlotStyle->Thick],Graphics[{PointSize[.02],Point[{3*t,3*t}]}]],{t,0,25},AnimationRunning->False,AnimationRepetitions->2]

Line parametrized by ⟨t+sin² t, t+sin² t⟩

Animate[Show[Plot[x,{x,0,30},PlotStyle->Thick],Graphics[{PointSize[.02],Point[{t+Sin[t]^2,t+Sin[t]^2}]}]],{t,0,25},AnimationRunning->False,AnimationRepetitions->2]