Tweening a mesh does not make it move?

That's because the variable bert is referring to the object inside ReplicatedStorage. When you clone the object, it still will refer to the original object.

Instead, create the Tween after the object has been cloned, and refer to that instead.

local RStorage = game:GetService("ReplicatedStorage")

local SummonBert = RStorage:WaitForChild("SummonBert",math.huge)

local bert = RStorage:FindFirstChild("bigcat")

local tweenService = game:GetService("TweenService")



local Info = TweenInfo.new(

    10  
)

local goal = {

    Position = Vector3.new(-36, 596, -15)

}

local MoveBert = tweenService:Create(bert, Info, goal)



MoveBert.Completed:connect(function()
    for i,v in pairs(game:GetService("Players"):GetPlayers()) do
        wait(3)
        v:Kick("Nothing remains in this world")
    end
end)


SummonBert.OnServerEvent:Connect(function(player)
    local bertClone = bert:Clone()
    bertClone.Parent = workspace
    local MoveBert = tweenService:Create(bertClone, Info, goal)
    print("Bert has entered workspace")
    wait(3)
    MoveBert:Play()

    print("OUR LORD AND SAVIOR, BERT HAS BEEN SUMMONED, ALL HAIL BERT")
end)