There are a few things wrong with what you wrote.
If you capture at the hosts, there will not be a VLAN tag on the frame. VLAN tags are used on trunks where there are multiple VLANs so that the network devices can tell which frames belong to which VLAN.
The switch does not broadcast unless the frame is a broadcast frame. A switch will have a MAC address table that has the MAC addresses from the frames' source addresses, and from which switch interfaces the source addresses were seen coming into the switch. The switch will look up a frame destination MAC address in the MAC address table and send the frame to the corresponding interface. If the switch doesn't find the destination MAC address in the table, it will flood the frame to all other switch interfaces in the same broadcast domain (VLAN).
A host will compare the destination IP address to the hosts's IP address and mask to see if it is in the same network. If the destination MAC address is in the same network, it will use ARP to discover the the destination host's MAC address, and use that as the destination MAC address for the frame.
If the destination host's IP address is in a different network, the host will use the MAC address of the configured gateway as the destination MAC address for the frame.
If the connection between the switch and the router is a trunk, both the switch and the router will tag any frames for non-native VLANs. The switch will not include a VLAN tag on the access ports.
Edit:
A router will have separate logical interfaces, on the same physical interface, for each VLAN. For a Cisco router, it looks something like this:
interface GigabitEthernet0/0
description Physical Interface
no shutdown
!
interface GigabitEthernet0/0.10
description VLAN 10 Interface
encapsulation dot1Q 10
ip address 10.10.10.1 255.255.255.0
no ip redirects
no ip unreachables
no ip proxy-arp
no shutdown
!
interface GigabitEthernet0/0.20
description VLAN 20 Interface
encapsulation dot1Q 20
ip address 10.20.20.1 255.255.255.0
no ip redirects
no ip unreachables
no ip proxy-arp
no shutdown
!
The encapsulation
command on the interface will direct incoming frames on the physical interface to the correct logical interface, and it will add the VLAN tag to any frame exiting the logical interface outbound through the physical interface.
A router will strip off and discard the layer-2 frame to get to the layer-3 packet. After switching the packet to the new interface, a router needs to build a new frame for the new interface.
The VLAN tag added to the frame by the switch on the trunk to the router will be lost when the frame is stripped. The router will add a tag for the new VLAN when it builds the new frame for the new interface. That is where the VLAN tag is changed.