Why do reactivity increases on moving from top to bottom in group 2 of modern periodic table?

that's because

group 1 (e.g Na, K) those tend to lose one electron to gain noble gas electron configuration.

they can achieve that by just losing one electron from their outer shell.

as you go down the group 1, element gets bigger in size, which means there is more space between nucleus (which is in center of atom) and electron of outer shell. the more far away they are the less attraction force between them.

so its easier for potassuim to lose one electron than for lithuim.

so that means potassium will easily give up 1 electron to react with non metal or other element therefore it is more reactive than lithuim

but in case of non metal, the opposite happens but simple to understand.

as you go down the group 7 (halogen- Cl, Br, I) element will get bigger therefore force between nucleus and outer electron is getting smaller. they have to gain 1 electron in order to fill the outer shell (to gain noble gas electron configuration.)

as florine is more smaller in size than clorine it is more reactive because florine has more tendency to pull extra electron from metal or other element towards its side. so it easily gain 1 electron to react.

particle   p is an electron and particle q is a neutron