Some random animations to start with

z=99/100;
Animate[Graphics[
  {Circle[{0,0},Sqrt[1-(z^2)+(((z t))^2)]],Circle[{0,0},z t]}
,PlotRange->{{-1,1},{-1,1}}],{t,0,1}]

Animate[Plot[
{Sqrt[1-z^2+(z t)^2-x^2],-Sqrt[1-z^2+(z t)^2-x^2],Sqrt[(z t)^2-x^2],-Sqrt[(z t)^2-x^2]}
,{x,-1,1},PlotRange->{Full,{-1,1}},PlotStyle->{Black}],{t,0,1}]

Animate[Plot3D[
{Sqrt[1-y^2+(y t)^2-x^2],-Sqrt[1-y^2+(y t)^2-x^2],Sqrt[(y t)^2-x^2],-Sqrt[(y t)^2-x^2]}
,{x,-1,1},{y,-1,1},PlotRange->{Full,Full,{-1,1}},BoxRatios->{1,1,1}],{t,0,1}]

A function that goes from 0 to 1 to 0, moving quickly through the middle stages

h=((-0.01`+1.01` (E^(1.02` (-4.5246279576875095`+9.049255915375017` x))))/(1+(E^(1.02` (-4.5246279576875095`+9.049255915375017` x)))));

tl=Join[{0,0,0,0,0},Table[h,{x,0,1,1/30}],{1,1,1,1,1},Table[h,{x,1,0,-1/30}]];

ListPlot[tl]

Here is the main animation:

T=Table[Plot3D[
{Sqrt[1-y^2+(y t)^2-x^2],-Sqrt[1-y^2+(y t)^2-x^2],Sqrt[(y t)^2-x^2],-Sqrt[(y t)^2-x^2]}
,{x,-1,1},{y,-1,1},PlotRange->{Full,Full,{-1,1}},BoxRatios->{1,1,1},PerformanceGoal->"Quality"],{t,tl}];

Export["cone sphere.gif",T]

Here is a picture of one of the slides

T[[20]]