Is there anything like Mouse.Move for UserInputService?

1. I'll just say that I don't know how you can do this with UserInputService, but mouse.Move already seems to work fine.

2. I tested out your script. Everything works well except for the movement of the targetPart. I don't know if that's why you wanted to use something other than mouse.Move. I used TweenService in your onMouseMove function (on line 43), and the movement was fixed.

3. When I let go of the part which is where InputEnded runs, the targetPart went back to its original position. That was fixed on line 53.

local UserInputService = game:GetService("UserInputService")

local mouse = game:GetService("Players").LocalPlayer:GetMouse()
mouse.TargetFilter = script.Parent -- I wrote script.Parent because it's in StarterCharacterScripts

local wasAnchored 

local wasCanCollide

local targetPart

local selectionBox

local mouseMove

UserInputService.InputBegan:Connect(function(input, gameProcessedEvent)
    if input.UserInputType == Enum.UserInputType.MouseButton1 and not gameProcessedEvent then
        targetPart = mouse.Target

        selectionBox = Instance.new("SelectionBox")
        selectionBox.Adornee = targetPart
        selectionBox.Parent = targetPart

        if targetPart.Anchored then
            wasAnchored = true
        else
            wasAnchored = false
        end

        if targetPart.CanCollide then
            wasCanCollide = true
        else
            wasCanCollide = false
        end

	local tweenService = game:GetService("TweenService")
	local player = game:GetService("Players").LocalPlayer

	local function onMouseMove()
		if not targetPart.Locked then
			targetPart.CanCollide = false
	        	targetPart.Anchored = true
    			local tween = tweenService:Create(targetPart, TweenInfo.new(0.4, Enum.EasingStyle.Sine, Enum.EasingDirection.Out, 0, false, 0), {CFrame = CFrame.new(mouse.Hit.p)})
    			tween:Play()
		end
	end
	mouseMove = mouse.Move:Connect(onMouseMove)
end
end)

UserInputService.InputEnded:Connect(function(input, gameProcessedEvent)
    if input.UserInputType == Enum.UserInputType.MouseButton1 and not gameProcessedEvent then
		targetPart.Position = mouse.Hit.p
        mouseMove:Disconnect()
        selectionBox:Destroy()

        if not wasAnchored then
            targetPart.Anchored = false
        end

        if wasCanCollide then
            targetPart.CanCollide = true
        end
    end
end)