To prevent players from acting like vultures, you might want to think about having the person who does the most damage being credited with the kill. Though this approach would most likely not work in a FFA environment. Granted you're talking about assists, so I'd imagine this is some semblance of a team game. Another thing I just thought of that would probably increase the cohesiveness of the team is completely eliminating the individual kill counter, and only displaying an overall team kill count. Suddenly the goal shifts from being I have to be the last one to hit this, or I have to snipe this kill, to I have to kill him and I'm glad this guy is helping me secure the kill for the team. If you implemented this approach, you could track an individual stat like damage done which is completely representative of your contribution to the team instead of only representing how good you are at getting the last hit.
Edit to further address your question:
Something you might want to keep in mind for calculating assists are metrics other than pure damage. For instance, say two pods are trading fire. If the two fire at equivalent rates, the result is dependent on who started shooting first/who is the most accurate. Assuming all factors are equal, add a healer into the mix. Now the player being healed with come out on top barring intervention by others. In my mind, the player doing the healing should get an assist. without him, the player doesn't get the kill. Other ideas to keep in mind are debuffs applied to the enemy and other buffs applied to the killer.